Physics <BR>Textbook Website: Puzzle solutions


PUZZLE SOLUTIONS

for
"New Century Senior Physics
- Knowledge, Processes and Reasoning"
- by Richard Walding, Greg Rapkins and Glenn Rossiter
Oxford University Press



This webpage has all the answers to the
Complex Reasoning Puzzles
in the margins of the second edition
New Century Senior Physics

Text box page 3
Complex Reasoning Puzzle
If you were transported in a time machine to an unknown date in Australian history, how could you work out the date?
ANSWER: Several ways: (a) you could look at the fauna (eg dinosaurs) and relate this to their extinction history; (b) you could look at the position of the Sun with respect to the constellations. This would give you some idea but because the "precession" cycle repeats itself, you’d have to know which billion years you’re in beforehand; (c) if you knew the changes in sea level you could work out a rough time, but only if you knew which global warming cycle you were in; (d) you could check which version of Windows your school was using and add 10 years on to that. Any other suggestions? Check if any sausage rolls at the tuckshop had gold watches for long service?

Text box page 3
Complex Reasoning Puzzle
The four compass directions North, East, South, West are derived from old foreign words. Can you match up the original meanings with the compass directions:
A – Indoeuropean wes = sun goes ‘down’; B - Italian nerto = ‘to the left’ as one faces the sun; C – German suntha = region in which the ‘sun’ appears in the northern hemisphere; D – Indoeuropean aus = sun ‘rises’.
ANSWER: A = west; B = north; C = south; D = east.

Text box page 4
Complex Reasoning Puzzle
You have two 100-page volumes of a dictionary on your shelf.
A worm eats its way from Volume 1 page 1 through to Volume 2 page 100. How many pages does it eat through?
ANSWER: None (or two at the most).

Text box page 5
Complex Reasoning Puzzle
Consider the Earth to have a circumference of 40 000 km and a ribbon to be put tightly around it. If you cut the ribbon and inserted a 30 cm piece, how far will the ribbon be from the Earth if it was evenly spaced?
ANSWER: 4.8 cm (the 40 000 km circumference makes no difference). Use C = 2p r and for a circumference of 30 cm, r works out to 4.8 cm. If you try it the long way you won’t get a significant difference.

Text box page 5
Complex Reasoning Puzzle
Quick now – is a physics lesson longer or shorter than a microcentury?
ANSWER: Shorter (well, presuming your lesson is less than 52 minutes). Forty minutes is long enough.

Textbox page 6
Complex Reasoning Puzzle
In the first paragraph of Charles Dickens’ The Pickwick Papers, he states that he was at the bottom of a deep well and could see the stars in the daytime. Aristotle made the same claim in On the Generation of Animals in 350 BC. Is this possible? Propose points for and against this idea.
ANSWER: No, you can see the stars. All you see is a brilliant white light. We asked a Cornish tin miner what he recalled seeing and he said "bright sunlight". So there!

Textbox page 6
Complex Reasoning Puzzle
What time is it at the South Pole.
ANSWER: They use Eastern Standard Time (Brisbane).

Text box page 7
Complex Reasoning Puzzle
Under the system of measurement adopted during the reign of Queen Elizabeth I:
1 mouthfull = 1 cubic inch
1 handfull = 2 mouthfull
1 jack = 2 handfull
1 jill = 2 jacks
1 cup = 2 jills
1 pint = 2 cups
1 quart = 2 pints
If 1 cubic inch = 14.7 mL, how many cups to 1 litre?
ANSWER: 1 L = 68 cubic inches = 68 mouthfull = 34 handfull = 17 jacks = 8.5 jills = 4.25 cups.
Let me tell you a story about jacks and jills. Remember the nursery rhyme about:

Jack and Jill went up the hill,
to fetch a pail of water.
Jack fell down and broke his crown,
and Jill came tumbling after.

In the mid-1600s, English king Charles I placed a tax on beer and spirits to raise money for his own pleasure. At that time drinks were sold by the jackpot which had the volume of one jack, and the jill. His subjects resented this new tax. They resented it even more when he reduced the size of the jack and jill to increase his revenue even further. Under his tough rule protests had to be disguised so the people made up the rhyme about jack and jill. The words about "jack and jill went up the hill" refers to the increase in price for a jack and jill of drink as the volume was decreased; "jack came down" is about the measure returning to its original size; "broke his crown" refers to Charles I being almost toppled from the throne as the people revolted; and "jill came tumbling after" means that the jill returned to the original size.

If you know what being patched with vinegar and brown paper means, please tell us.
Update: Physics student Amy of Brisbane writes: My Mum says that "Vinegar and Brown Paper" was an old remedy by the English for healing that was used as frequently then as band-aids are used now. I also found out from my own research that the four line ending to the poem was actually added later on to make the rhyme more suitable for children to sing, and hence the lyrics above were never in the original version.


Text box page 8
Complex Reasoning Puzzle
Question about Schneider vs Taylor.
ANSWER: Back in the 13th century, surnames were first used and these were related to occupations. People who made clothes (tailors) adopted the surname Tailor or Taylor (Schneider in German); people who were blacksmiths took the name Smith (or Schmidt in German). Smiths were big strong men whereas Tailors were generally smaller and not so strong. The average mass of Smiths was higher than for Tailor (by about 10 kg back then). This difference is now down to about 3 kg. People with the surname King – don’t think your ancestors were kings. During the May (spring) Festival, people often dressed up as kings, queens, soldiers etc for fun and sometimes took their surname from this.

Text box page 12
Complex Reasoning Puzzle
Number sense question.
ANSWER: The hypothesis being tested was that the further a number is away from a given number the easier it is to say whether it is greater or less than the given number. What's more - it's true that it is harder when the numbers are close together. Even scientists take longer to figure it out.

Text box page 12
Complex Reasoning Puzzle
Famous biologist Charles Darwin described the size of a canary finch…etc. Convert the original measurement to centimetres using the correct number of significant figures.
ANSWER: A few possibilities: (a) his ruler was in 64ths and this was the usual way to express measurements; (b) 3 ½ inches has only 2 significant figures and Darwin wanted to be more precise so he used 3 32/64 which has 3 sf. In centimetres, the measurement is 3 32/64 x 2.54 = 8.89 cm (to 3 sf).

Text box page 12
Complex Reasoning Puzzle
In the English translation of a manual on violin playing by the great Hungarian-German teacher Carl Flesch, it told budding violinists to "lift your fingers 0.3937 inches from the fingerboard". Why is this funny? What do you suppose the original measurement was? Rewrite the inches measurement with the correct number of significant figures.
ANSWER: It is funny (to us anyway – maybe not to you) because it has a ridiculous number of significant figures. A distance of 0.3937 inches = 0.3937 x 2.54 cm (= 1 cm). Carl Flesch would have written "lift your fingers 1 cm from the fingerboard" which has one significant figure. In inches this would be 0.4 inches.

Text box page 12
Complex Reasoning Puzzle
In a shop in North Walsham, Norfolk, the height restriction to its carpark is written as 2300 mm. Is there anything wrong with this? Explain!
ANSWER: They obviously meant the measurement to be 2.3 m and so strictly speaking it is correct. But to most people it would look like a car 2299 mm high would fit but not one 2301 mm high. This is plainly absurd. They should have written 2.3 m.

Text box page 13
Complex Reasoning Puzzle
Muttabuttasaurus question.
The age estimate 30 million years would be 30 million give or take a few million (probably 25-35 million) seeing it has only one significant figure. A change of 20 years would not make this number any different. To one significant figure it would still be 30 million years. (We think the real age of the fossil in the Queensland Museum is 100 million years old – they’d died out about 60 million years ago).

Text box page 13
Complex Reasoning Puzzle
19 is about 20.
ANSWER: 20 has one significant figure so really could mean anything from 15 to 25. The number 19 has 2 sf so could only mean something from 18.5 to 19.5.

Text box page 13
Complex Reasoning Puzzle
Tesco lemonade question….
ANSWER: The 8 kJ per 100 mL may have been 8.4 kJ rounded off to 1 sf so that two lots of 8.4 = 16.8 kJ which is 17 kJ to 1 sf. Just drink the 100 mL size and you won't get as fat.

Text box page 39
Complex Reasoning Puzzle
A car travels from A to B at an average speed of 100 km/h and returns at 60 km/h. What is the average speed for the journey?
ANSWER: 75 km/h. This is an old trap. Assume the distance between A and B was 600 km. It would take 6 hours to go there and 10 hours to come back. That’s a total of 16 h for a 1200 km trip. Hence 1200/16 = 75 km/h. Simple huh?

Text box page 40
Complex Reasoning Puzzle
A column of troops 3 km long is marching along a road. An officer rides from the rear to the head of the column and back once, and he reaches the rear of the column just as an advance of 4 km has been made from where he first left. How far did he ride?
ANSWER: 8 km (see working below):

1. Let the officer on the horse be "O"
2. Let the rider at the rear of the column be “R"
3. The velocity of the officer to the ground (vOG) = velocity of officer to rear (vOR) plus velocity of rear with respect to ground (vRG). That is, vOG = vOR + vRG
4. Let time for O to reach head of column be t1. Hence t1 = s1/vOG
5. In this time R has moved a distance s2. Hence t1 = s2/vRG
6. Both are equal to t1, hence: s1/vOG = s2/vRG
7. But s1= 3 + s2 thus (3 + s2)/vOG = s2/vRG
8. Rearranging:(3 + s2)/s2 = vOG/vRG
9 Let the distance O travels from the head of the column to the rear be s3
10. Let the time for O to travel from head of column back to rear = t2 (= s3/vOG)
11. In this time R has moved a distance s4 in a time t2 (= s4/vRG)
12. But s4 = 4 - s2
13. Substituting into equation in line 11: t2 = (4 - s2)/vRG
14. But s4 = 3 - s3 = 4 - s2. Hence s3 = s2 -1.
15. Substituting into line 10: t2 = s3/vOG = (s2 - 1)/vOG
16. Hence t2 = (s2 - 1)/vOG = (4 - s2)/vRG
17. Rearranging: vOG/vRG = (s2 - 1)/(4 - s2), but this equals (3 + s2)/s2 from line 8.
18. Hence (s2 - 1)/(4 - s2) = (3 + s2)/s2
19. Rearranging: (s2 - 1)s2 = (3 + s2)(4 - s2)
20. s22 - s2 = 12 - 3s2 + 4s2 - s22
21. 2s22 - 2s2 -12 = 0
22. Simplify: s22 - s2 - 6 = 0
23. Factorise: (s2 - 3)(s2 + 2) = 0. Hence s2 = 3 (ignore -2)
24. If s2 = 3, then s1 = 6, s3 = 2, s4 = 1
THUS - the officer travels s1 + s3 = 6 + 2 = 8 km
By the way, the velocity of the officer and column can be found:
25. Substituting into equation 7: vRG = vOG/2
26. But vOG = vOR + vRG, hence 2vRG = vOR + vRG, thus vRG = vOR
27. In the total time of t1 + t2, R has traveled 4 km, hence: vRG = 4/(t1+t2)
28. Substituting for t1 (equation 4) and t2 (equation 15): (using actual values of s1 and s3):
vRG = 4 /((6/vOG) + 2/vOG)) = 8/vOG
29. Hence vRG = 8/vOG = vOG/2 (from equation 25)
30. vOG2 = 16, hence vOG = 4, and vRG = 2.
The officer travels at 4 km/h and the column travels at 2 km/h.
The times are t1 = 1.5 hr, t2 = 0.5 hr. Total time 2 hours.

Text box page 40
Complex Reasoning Puzzle
A man goes from A to B at 30 km/h. How fast must he return to average 60 km/h for the whole trip?
ANSWER: Impossible. He’d have to return at an infinite speed (that is – in zero time). If you said 90 km/h then all we can say is "sucked in".

Text box page 41
Complex Reasoning Puzzle
A boy is carried up an escalator in 1 minute. He can walk up a stationary escalator in 3 minutes. How long will it take him to walk up a moving escalator?
ANSWER: Let the distance up the escalator be x metres.
The velocity of the boy with respect to the escalator vBE = x/3 metres/minute;
The velocity of the escalator with respect to the ground vEG = x/1 metres/minute;
Let the velocity of the boy with respect to the ground be vBG (to be found).
Hence: vBG = vBE + vEG = x/3 + x/1 = 4x/3 metres/minute;
But vBG = x/t, hence x/t = 4x/3; thus t = ¾ minute.
The question is hypothetical - the boy would probably be arm-wrestled to the ground by the security men and chucked out of the shopping centre.


Text box page 42
Complex Reasoning Puzzle
If you put a row of coins on a 1 metre ruler which has one end on the ground and let the other end fall – which coins will stay on the ruler and which ones will be left behind?
ANSWER: All the ones up to the centre will stay put. When the ruler falls, the centre of gravity will fall with an acceleration of 9.8 m s-2 whereas the end touching the table will stay at rest and the free end will accelerate faster than 9.8 m s-2. Try it.

Text box page 43
Complex Reasoning Puzzle
If a flea was as big as a man, how high could it jump (approximately)?
ANSWER: A flea can jump maybe 30 cm but humans are about 1800 mm high and a flea is about 1/10 mm high (we guess). So we’re 18000 times as high as a flea so if a flea can normally jump 30 cm, a human-sized flea could jump 18000 x 30 cm = 5.4 km. Now that’s high.

Text box page 60
Complex Reasoning Puzzle
Run or walk in the rain?
ANSWER: The number of raindrops that strike your body is made up of two components: (i) the number of drops that fall on you head and shoulders (DHS) and (ii) the number that you collect on the front of your body as you sweep through the volume of air containing the falling drops (DSV). The total number of drops on your body (DT) = DHS + DSV. The DHS is independent of your speed – you always sweep out the same volume of air whether you run or walk. The DHS however, depends on how long you are in the rain. The faster you run the fewer drops strike your head. So – in vertical rain – run as fast as you can. Rain at an angle is too complex. Wear a raincoat and run.

Text box page 63
Complex Reasoning Puzzle
A passenger on a train travelling at 60 km/h observes that it requires 4 s for another train 100 m long to pass her by. What is the speed of the second train?
ANSWER: 30 km/h
Let train A be travelling at 60 km/h with respect to the ground (= vAG);
The velocity of train B with respect to train A is 100m/4s = 25 m/s or 90 km/h;
Because train B is travelling in the opposite direction to train A: vBA = -90 km/h or vAB = 90 km/h;
vAB = vAG + vGB
vAB = vAG + -vBG
90 = 60 + -vBG
-vBG = 30
hence vBG = -30 km/h

Text box page 76
Complex Reasoning Puzzle
A fan blows a cart with a sail attached.
If the fan and the sail are on the same cart what happens. Explain why.
ANSWER: The air pushes the sail one way but the fan pushes against the air (Newton’s third law) the other way. So they sit there doing nothing. Zilch!

Text box page 76
Complex Reasoning Puzzle
Issac Newton’s early childhood was marked by rejection and hatred. He has been ranked as the second most influential person in the world (influential not important). Develop an argument for who might be 1st and 3rd ?
ANSWER: In Michael Hart’s book "The 100", subtitled "The most influential persons in history", he lists Mohammad first, and Jesus third. Check it out in the bookstore (published by Simon and Schuster, 1999). Don’t email us with your complaints.

Text box page 79
Complex Reasoning Puzzle
(a) If you took a beaker of wet sand from the beach, would it weigh more or less than the same volume of dry sand?
(b) Stand on some wet sand at the beach and it goes dry around your feet. Why is this?
ANSWER: (a) More (but we’re not game to check).
(b) Because it is disturbed from its closest packing arrangement by the weight of your body. The grains are forced into a less compact arrangement and puffs outwards and looks dry.

Text box page 80
Complex Reasoning Puzzle
Newton and science.
ANSWER: We think Newton invented them. Laws only exist because they have been created by someone. Radioactivity gets discovered, the laws of radioactive decay get invented. What about laws that make you wear a helmet when riding a bicycle? Were they discovered by Mr Stackhat?
 

Text box page 83
Complex Reasoning Puzzle
In the summer of 1693, John Bernoulli posed the following problem which still hadn’t been solved 6 months later. The day Newton heard it he solved it. The problem: you have three paths for a ball to roll down – which is the fastest?
ANSWER: The cycloid (middle) path is similar to the path traced out by a point on the rim of a wheel as it rolls along a horizontal surface. It is the path of least time.

Text box page 83
Complex Reasoning Puzzle
Armstrong’s weight question.
ANSWER: On Earth, he would weigh 3 x 1000 ounces = 3000/16 pounds = 187.5 lb. This is equivalent to a mass of 187.5/2.2 = 85.2 kg.

Text box page 85
Complex Reasoning Puzzle
A man comes up to a bridge which can just support his weight and 1 of the 4 balls he is carrying. He decides to juggle as he crosses so that only 1 ball will be in his hands at any one time. What do you think of his solution?
ANSWER: Terrible. Although three are in the air, he has imparted a force to the 4th ball greater than its weight (to make it accelerate upwards). This is greater than the bridge can stand.

Text box page 86
Complex Reasoning Puzzle
Throwing a ball while parachuting.
ANSWER: the drag on a tennis ball is more than on a skydiver in free-fall (see page 87 for some terminal velocities) so the ball would appear to fly upwards as they fell. Once they opened their parachutes the drag on the ball would be less than on them (see table, page 87) so the ball would start to fall downwards towards them.

Text box page 87
Complex Reasoning Puzzle
Imagine you were to drop a book which has a paper napkin resting on the top. How will they both fall?
ANSWER: The napkin stays on the top of the book. There is a decrease in pressure on the top of the book which keeps the paper napkin in place. It like cyclists being ‘sucked’ along behind a truck. Don’t try it.

Text box page 87
Complex Reasoning Puzzle
If you dropped a marble, a big styrofoam ball and a small one (from a bean bag) together from chest height, two hit the ground at the same time. Which two and would they be faster or slower than the other one? Why? Try it and see!
ANSWER: The marble and the small ball hit together. There is proportionally more air resistance for the big styrofoam ball than for the small one (air resistance is proportional to surface area which varies with the square of the radius). The big ball is slower.

Text box page 87
Complex Reasoning Puzzle
If the objective of a parachute is to slow the descent of a falling object in air, why do parachutes have a hole (the apex vent) in the top allowing air to escape. In World War II they didn’t have an apex vent and they swung like pendulums as they descended (watch an old war movie – you’ll see). What is the physics behind this?
ANSWER: The air has to escape from the parachute somehow so if there is no apex vent it tilts to the side to let it out. Then it tilts the other way to let it out the other side.

Text box page 90
Complex Reasoning Puzzle
(a) Which box requires the smaller force to lift?
(b) Which box requires less work to raise 1 metre?
ANSWER: The top box requires less force.
They both need the same amount of work to lift them 1 metre.
Why? Because in the top figure when the rope is pulled 1 metre the box only lifts up 0.5 m, This is called "double purchase". In the bottom figure (single purchase) the box rises 1 metre when the rope is pulled 1 metre. Hence in the top figure, half the force is required over twice the rope distance to move it compared to the bottom figure. The joy of pulleys.

Text box page 90
Complex Reasoning Puzzle
A monkey has the same mass as a box. He climbs a rope. Who will reach the pulley first?
ANSWER: They both reach it together. As the monkey pulls down on the rope and accelerates himself upwards a bit, the tension in the rope increases. This will pull the box up with the same acceleration and will move the same distance up the rope in the same time. As the monkey pulls again, the same thing happens so they both arrive at the top together. If it was a spider monkey, he'd probably steal your sunglasses.

Text box page 92
Complex Reasoning Puzzle
Motorcycle tyres come in two main types – sport and touring. Sport tyres are v-shaped, touring are rounded. The figure below shows the profiles of Dunlop’s mega-successful D207GP race tyre which has been cleaning-up in supersport competitions for some time now.
(a) Plot a graph of the lean angle (from 5° to 50° in 5° increments) against the cosec of the lean angle. Note when q = 30° , cosec 30° = 1/sin 30° = 2. A computer spreadsheet may speed things up. How does the curve compare to the groove shape in the D207GP tyre?
(b) As riders using the v-shaped tyre go into a corner they flip the bike on to its side rather than doing a uniform lean. Why is this? The upper limit of contact between the tyre and the road is called the ‘hero line’. What do you suspect this means?
(c) Cruising bikes like the Harley Davidson have tyres with a flat profile. This must impair their cornering at speed so why do they have it?
ANSWER: (a) it has the same shape (see figure below).

(b) to get maximum contact area straight away.
(c) a measure of what sort of hero you are is how far over you can lean. There’s a fine line between being a hero and an idiot.

Text box page 93
Complex Reasoning Puzzle
A piece of pine dowel is placed in an electric drill and rotated against a piece of hardwood. Which will catch on fire first? Try it.
ANSWER: The pine. We tried it. The fibres in hardwood are more tightly packed and act as an insulator against the heat spreading.

Text box page 93
Complex Reasoning Puzzle
A block of wood is placed on top of a smooth cylinder.
When the block gets to 45° it slides off. If a ball was used and allowed to roll down the surface, would it fall off at a bigger or smaller angle than 45° ? Justify your answer.
ANSWER: The rolling ball will stay on longer. If you can do the mathematics (we can’t), the sliding body leaves at 48° and the rolling body leaves at 54° . We are told that the sliding block leaves when the cosine of the angle is 2/3 and the rolling ball leaves when the cosine is 10/17. Simple huh?

Text box page 93
Complex Reasoning Puzzle
A cubical block of mass 10 kg is placed on surface where m = 1.0. Where about on the block would you have to push it so that it was on the verge of tipping and sliding?
ANSWER: Push it in the middle of the vertical face. This will make it be on the point of tipping and sliding. If you push it higher it will tip; lower and it will slide.
 

Text box page 96
Complex Reasoning Puzzle
A wooden block is put on an electronic balance and it reads 100 g (weight = 1 N). A string is attached by a spring balanced and a force of 0.4 N applied at an angle of 45° . What will the balance read? Try it to check.
ANSWER: The vertical component of the force applied is 0.4 sin 45° = 0.28 N;
Hence the balance registers the weight less the upward pull (1 – 0.28 = 0.72 N);
The balance would then read 72 grams. We do this demonstration in class and it works like a charm.

Text box page 97
Complex Reasoning Puzzle
A 10 kg block …etc.
ANSWER: Push on the 5 kg block.

Solution: Consider the push on the 5 kg block:
Total friction = Ff of 10 kg block + Ff of 5 kg block
= m x 100 + m (mg + mg sin 30° ) = 50 + 0.5(50 + 50 x 0.5) = 87.5 N
Consider the push on the 10 kg block:
Total friction = Ff on 10 kg block + Ff on 5 kg block
= m (mg + mg sin30° ) + m x 50 = 0.5(mg + mg sin30° ) + 0.5 x 50 = 75 + 25 = 100 N

Text box page 107
Complex Reasoning Puzzle
The following graphs show how the range and altitude of a projectile changes with elevation angle in the presence of air. Plot a graph of maximum altitude versus elevation angle and predict maximum altitude for angle of 90° . Should the graph pass through the origin (0,0). Why?
ANSWER: It should pass through the origin. If you shoot it into the ground it will go zero metres. The graphs are as follows:


Textbox page 108
Complex Reasoning Puzzle
A really hard one! A cannonball is fired and after travelling 5 m horizontally, it has reached half its maximum height. After what horizontal distance will it land?
ANSWER: 33.3 metres (or 34.14 to be more exact)
This is a difficult, difficult problem and will defy the best physics tutor around.
I (RW) tried several ways but you might find a quicker way.
ONE METHOD:
You can work out what fraction of the time a projectile spends in the bottom half of its journey (it doesn’t matter what the launch speed is, the fraction will always be the same). I worked out that it spends 30% of its time in the bottom half. Hence, it travels 5 m horizontally in 30% of its time to get to maximum height. Therefore, it will travel 16.65 m horizontally when it reaches its top of flight. By the time it lands again it will have travelled 33.3 m.

Initial velocity = v

Alternative solution provided by Steve Shellshear (you’ve really got to get out more Steve).

 

Initial velocity = v

Angle of projection = q

Initial vertical velocity component uv= vsinq

Initial horizontal velocity component vH= vcosq

 

Travel 5 m (sH) horizontally in time t:

Rearranging vH = sH/t gives sH = vH t = vcosq

Substituting: 5 = vcosq t

Rearranging: t = 5/ vcosq  (EQUATION 1)

 

Travel vertically half maximum height (sv = hmax/2) in time t:

Using sv = uvt + ½at2

hmax/2 = vsinq t + -5t2 (EQN 2)

 

Substitute Eq 1 into Eq 2:

hmax/2 = vsinq 5/ vcosq + -5(5/ vcosq)2

Multiply both sides by 2:

hmax/2 = 10sinq/cosq + -250/(vcosq)2 (EQN 3)

 

Determine hmax using vv2 = uv2 + 2as

0 = (vsinq)2 – 20 hmax

Rearranging:

20hmax = (vsinq)2

Divide both sides by 20:

hmax = (vsinq)2/20  (EQN 4)

 

Substitute EQN 4 into EQN 3:

(vsinq)2/20  = 10sinq/cosq + -250/(vcosq)2

Multiply both sides by 20 (vcosq)2

(vsinq)2(vcosq)2 = 200sinq (vcosq)2/cosq – 250 x 20

v4 (sinqcosq)2 = 200sinqcosq v2 - 5000

Prepare for identity conversion of 2 sinqcosq = sin2q

v4 (2sinqcosq)2/4 = 100 x 2sinqcosq v2 - 5000

Convert and multiply both sides by 4:

v4 (sin2q)2 = 400 v2 sin 2q - 20000

Subtract 400v2sin2q from, and add 20000 to both sides:

v4 (sin2q)2 – 400v2sin2q + 20000 = 0 (Equation 5)

 

Calculate the range using time t

Vertically, using sv = uvt + ½at2

0 = vsinq t – 5t2

Add 5t2 and subtract vsinqt from both sides:

5t2 - vsinqt = 0

Divide both sides by 5:

t2 - vsinqt/5 = 0

Factorise:

T(t-vsinq/5) = 0

Therefore, t = o t = vsinq/5

We are interested in t = vsinq/5  (EQN 6)

 

Horizontal range, R:

R = vcosqt (EQN 7)

R = vcosq vsinq/5

Convert to sin 2q

R = v2 (2sinq cosq)/10

R = v2 sin2q/10  (EQN 8)

Substitute EQN 8 into EQN 5

100R2 – 4000R + 20000 = 0

Divide both sides by 100

R2 – 40R + 200 = 0

Using quadratic formula:

R = (40 ± Ö(402 – 4 x 1 x 200)/(2 x 1)

R = (40 ± 28.284)/2

R = 34.14m or 5.86m

 

Thanks Steve.


Alternative solution by John Liu.
 The Cartesian form of the parabolic equation for projectiles is
y= xtanA-g(xsecA)^2/2v^2,
where x is horizontal distance, A angle of projection, g acceleration due to gravity (9.8 ms^-2), and v speed of projection.(^ is "to the power of-")
From the question, at x=5 the particle reaches half the maximum height, or
y(max)/2=5tanA-(25g(secA)^2)/2v^2. Multiply through by 2:
y(max)=10tanA-(25g(secA)^2)/v^2
On the other hand, by what we learn in year 12 mathematics, y(max) is ((vsinA)^2)/2g.
Equating this to the y(max) given above,
10tanA-(25g(secA)^2)/v^2=((vsinA)^2)/2g. Multiplying through by 2gv^2,
20gv^2tanA-50g^2(secA)^2=v^4(sinA)^2. Rearranging terms,
v^4(sinA)^2-20gv^2tanA+50g^2(secA)^2=0.
The range, also from year 12 mathematics, is v^2sin2A/g. Since the equation above is a quadratic in v^2, solve it for v^2 gives
v^2=(20gtanA+/-Sqrt(400g^2(tanA)^2-200g^2(tanA)^2))/(2(sinA)^2), where +/- means plus or minus, Sqrt means square root of.
This is
(20gtanA+/-10gtanASqrt2)/(2(sinA)^2), or
(20gsinA+/-10gsinASqrt2)/(2sinAcosA), or
(10+/-5Sqrt2)g/sin2A.
Substitute this value into range, we get
v^2sin2A/g=(20+/-10Sqrt2)g/sin2A x sin2A/g=(20+/-10Sqrt2). (The x in this statement is "multiplied by", not a variable.)
Because 20-10Sqrt2=5.86, this indictates the projectile fell to the ground less than one meter after taking 5 meters to reach half the maximum height! Therefore it cannot be the range the projectile travelled. So we must take 20+10Sqrt2=34.14 as the range.
Answer: the range travelled by the projectile is 34.14 meters.




Text box page 111
Complex Reasoning Puzzle
When a wheel rolls along, is any point at rest?
ANSWER: Yes – where it makes contact.

Text box page 112
Complex Reasoning Puzzle
Some coins were placed on a turntable and then it was turned on. What do you predict will happen?
ANSWER: The outer ones will fly off because they have a higher velocity and therefore a larger centripetal force is needed to keep them travelling in a circle. The friction provides the centripetal force but if the centripetal force needed exceeds friction they will fly off.

Text box page 112
Complex Reasoning Puzzle
How many revolutions will coin A do while rotating around coin B? Try it. You’ll be surprised.
ANSWER: Two

Text box page 112
Complex Reasoning Puzzle
Government steamer Relief and it’s whirling windscreen.
ANSWER: The water was flung off by the spinning disk. Car wipers wouldn’t be quick enough to keep the screen clear. One problem with a spinning disk is getting a good seal so water doesn’t leak in.

Text box page 113
Complex Reasoning Puzzle
A wheel is rolling along with constant speed and a lump of mud is thrown off its hindmost point. Will it touch the wheel again?
ANSWER: Yes. It follows a parabola.

Text box page 119
Complex Reasoning Puzzle
A candle with a nail through the middle is supported on two glasses and lit at both ends. How could you check if the resulting motion is SHM or just periodic?
ANSWER: Hmmm! This is a tough one. Perhaps you could videotape it or set a weighted spring oscillating up-and-down beside it and adjust the mass to see if you can get its motion to match the candle.

Text box page 120
Complex Reasoning Puzzle
A pendulum has a bucket for the bob and it is half-filled with water. When it freezes, predict what happens to the period of the pendulum?
ANSWER: Period decreases because the water expands upwards in the bucket as it freezes and the centre of gravity of the bucket is higher than before. This results in a faster pendulum.

Text box page 122
Complex Reasoning Puzzle
Two side-by-side pendulums are oscillating. One has a period of 6 s and the other a period of 7 s. If the bobs are touching at one time, how much longer must you wait until they come together again.
ANSWER: 42 seconds.

Text box page 122
Complex Reasoning Puzzle
How to make a swing go higher.
ANSWER: Your answer should be about why you lean back just as you get to the maximum amplitude backwards. When you lean back you pull on the rope and this has two components: one component (vertical) lifts you up out of the seat (no use); the other (horizontal) pulls you forward and increases your speed.
You could also analyse this movement in terms of the conversion of GPE to KE. The higher your centre of gravity when you are at maximum amplitude, the great your GPE and hence there is more energy to convert to KE. To get a high centre of gravity you could stand up or kick your legs high in the air. You should try to stay low as possible at the bottom of the swing so you tuck your lrgs under the seat. How do you slow down without skidding your feet? Do you just do the reverse?

Text box page 131
Complex Reasoning Puzzle
Newton said rationem vero harum Gravitatis propietartum ex phenomenis nondrum potui decucere "But I have not been able to discover the reason for this property of gravitation from the phenomena". What did he mean?
ANSWER: He could work out the mathematical relationships but could not work out the cause of the attraction.

Text box page 131
Complex Reasoning Puzzle
Two spherical drops of mercury are resting on a frictionless surface. The only force between them is that of gravitation. What would you need to know to be able to calculate how much time it would take for them to touch?
ANSWER: Hmmm! Friction. How would we know, we’re only high school physics teachers.

Text box page 133
Complex Reasoning Puzzle
Two balls are rolled around the inside surface and outside surface of a pipe whose diameter is 3 times the diameter of the balls. How many times will the outside one turn to complete one rotation of the pipe compared to the inside one?
ANSWER: Twice. The outer one rotates 4 times while the inner one rotates twice. The circumference of the circles traced out by the centre of each ball is found by 2p r. For the outer ball, r = 2 x diameter of the ball; for the inner one, r = diameter of the ball.

Text box page 135
Complex Reasoning Puzzle
How far would you have to travel horizontally out from the Earth for your altitude to be 1 km?
ANSWER: 113 km. The distance AC = 6.38 x 106 m (=EC); AB = 1000 m; hence BC = 6.379 x 106 m.
Using Pythagoras’ Theorem, BE = 113 000 m.



Text box page 137
Complex Reasoning Puzzle
If the Earth stopped rotating, how much would a 60 kg (590 N) person weigh? What would their bathroom scales read?
ANSWER: About 60.4 kg (592 N)
Apparent weight (FN) = FW - FC = 590 – (4p 2 r m / T2)
590 = FN – 2
FN = 592 N; m = F/g = 592/9.8 = 60.4 kg

Text box page 141
Complex Reasoning Puzzle
What would you hear 1 km from the Big Bang.
ANSWER: (a) the easy one: there is no air so there would be no sound; (b) the one we said you’d miss: at the time of the Big Bang, there was no ‘outside’ of the Big Bang. Space was being created by the expansion of the explosion. So you couldn’t have been 1 km outside. Question: could God have been 1 km outside the Big Bang?

Text box page 147
Complex Reasoning Puzzle
Which of the following ball bearings will fall slowest?
ANSWER: The one in water at -10° C. The water will be frozen.

Text box page 147
Complex Reasoning Puzzle
You can exert a force of 250 N with your incisor teeth and 1220 N with your molars. Which do you estimate to produce the higher pressure? Note: your front incisors are about 8 mm x 0.2 mm and your molars are about 8 mm x 8 mm.
ANSWER: Incisors have higher pressure.
P (incisors) = F/A = 250/(0.008 x 0.0002) = 1.56 x 108 Pa
P (molars) = F/A = 1220/(0.008 x 0.008) = 1.91 x 107 Pa

Text box page 148
Complex Reasoning Puzzle
Four car tyres are inflated to the same pressure. One wheel is jacked up. How does this change the pressure in the jacked-up tyre and the other three?
ANSWER: Increase pressure in jacked-up tyre and decrease it in other three.

Text box page 149
Complex Reasoning Puzzle
A fluorescent light tube is stood upright in a bucket of water and a small hole is cut in the tube underwater with a triangular file. The gas pressure inside the tube is 330 Pa. To what height do you estimate the water will rise? Try it but first think of how you will dispose of the water-filled tube.
ANSWER: About 4 cm from the top.
Solution: A tube is 1.2 m long and assume the water rushes in and rises to a height "d" metres from the top. The column of water and the gas at the top of the tube has to be supported by the outside atmospheric pressure of 101300 Pa. The pressure in the column = r gh = 1000 x g x (1.2 - d). To this we have to add the pressure of the gas. Originally it was 330 Pa but it is now compressed. Assuming the cross-sectional area of the tube is "A": using Boyle's Law: P1V1 = P2V2 we get 330 x (1.2 A) = P2 (d A); P2 = 330 x 1.2/d.
Hence: Atmospheric pressure (101300 Pa) = 1000 x 10 x (1.2 - d) + 330 x 1.2/d
101300 = 12000 - 10000d + 3960/d
101300d = 12000d - 10000d2 + 3960
10000d2 + 89300d - 3960 = 0 (in quadratic equation form)
Solve for d and you get d = 0.044 m (= 4.4 cm). Voila!
To get the water out of the tube - put it in a wheelie bin and tap the bottom of the tube with a hammer. You should wear gloves. Alternatively, leave it standing in the bin for the cleaner.

Text box page 149
Complex Reasoning Puzzle
An empty soft-drink can has some water in it and is boiled over a bunsen burner. It is quickly inverted and stood up in a tray of cool water. What do you predict will happen? Try it – but use tongs.
ANSWER: It will collapse. As the steam condenses, the internal pressure is reduced so the pressure of the atmosphere crushes the can.

Text box page 149
Complex Reasoning Puzzle
A piece of burning paper is placed in a conical flask and a boiled egg placed in the top. You can imagine what happens. But how do you reverse the process without touching the egg – now that’s where the physics is needed.
ANSWER: Invert the bottle so that the egg sits in the neck. Put your mouth over the opening and blow hard. The air rushes past the egg and the high pressure inside the bottle should then push the egg out. It works in theory but we’re not game to try it in case we choke and have to give up teaching.

Text box page 151
Complex Reasoning Puzzle
The sphygmomanometer was invented by René Laënnec in 1816. He used a rolled up tube of paper to listen to a very fat patient. Why isn’t the cuff wrapped around the lower part of your arm. Why wouldn’t you measure the pressure in that big artery in your neck?
ANSWER: Because you’d cut off blood to your brain and pass out or die. At least you wouldn’t have to sit the Core Skills Test.

Text box page 153
Complex Reasoning Puzzle
I was in the swimming pool ….etc
ANSWER: It is true that the apparent weight of an object (the steel spanner) is less in water than in air because of the upthrust; but that is not the point of the question. The question said the spanner felt heavier (not lighter). Your hand also experiences an upthrust as it displaces water; but it experiences a much greater upthrust than the spanner because it displaces a greater amount of water (perhaps twice as much). So you hand is being pushed upwards against the spanner and to you the spanner feels heavier than it should. If you got this right, Archimedes would have been proud of you.
For example: a 500 g steel spanner has a volume of 64 cm3. It’s weight is 5 N and the upthrust (= weight of water displaced) = 0.64 N; hence it’s apparent weight = 4.36 N. In air, your hand would feel a weight of 5 N but in water it should only feel 4.36 N. However, you hand has an upthrust as well. If your hand had a volume of 500 cm3, 500 g of water would be displaced and your hand would experience an upthrust of 5 N that you wouldn’t have in air. The apparent weight of the spanner would then be 9.36 N to your hand. This is more than the 4.36 N you expected.

Text box page 154
Complex Reasoning Puzzle
Density of some fluids (in g cm-3)
Petrol 0.68
Alcohol 0.81
Water 1.00
Seawater 1.03
Sugar syrup 40% 1.15
Mercury 13.6
1. What would have the greater density: (a) low alcohol beer or normal beer; (b) Coke or Diet Coke?
ANSWER: Low alcohol beer; Diet Coke.
2. Cans of Coke and Diet Coke were put in a tub of water. Will they both float, sink or one float, one sink?
ANSWER: They both sink but the Diet Coke is obviously more buoyant than the regular Coke. In the USA, the gas space in the can is just a bit more and in fact the Diet Coke floats over there. Well it did a few years ago.

Text box page 154
Test your understanding.
A cube of brass…etc.
ANSWER: 58 mL; 122 mL. If you said 74 mL you were sucked in!

Text box page 155
Complex Reasoning Puzzle
The density of salt water is greater than that of fresh water. So, swimming in salt water should be faster as your body floats higher and therefore there is less friction. True or false, and why?
ANSWER: No, on the contrary – you can swim faster underwater because this allows more efficient transfer of momentum to the water (which creates forward thrust) and because you don’t waste energy splashing water. Splashing and turbulence at the surface slows swimmers down because it increases drag. It is a misconception that the higher you are in the water, the faster you can swim. That’s why Olympic rules forbid swimmers travelling more than a certain distance (15 m) underwater or taking more than one full kick and one arm pull in breaststroke. You’d also think that any increase in buoyancy with saltwater would be offset by increased viscosity.

Text box page 159
Complex Reasoning Puzzle
A plastic soft drink bottle is half full of water. A small piece of cork is held just under the surface by a piece of string glued to the bottom. The bottle is slid to the right. Which way does the cork move relative to the bottle – forward, backward, sideways, no movement? You’ll be shocked if you try.
ANSWER: It moves the same way as the bottle. Look at the sloping surface of the water for a clue as to why. The cork moves to the part of the surface where it can float and have zero nett forces on it along the surface.

Text box page 160
Complex Reasoning Puzzle
A hollow steel ball….etc
ANSWER: This time we get 0.187 cm.
Let final radius of ball be r; and radius of inside is 4 cm.
Vsteel = 4/3 p r3 - 4/3 p 43 = 4/3 p (r3 – 43)
msteel = 4/3 p (r3 – 43) x 7.8 grams;
Vball = 4/3 p r3
When density = 1 it just floats; D = m/V, hence:
1 = 4/3 p (r3 – 43) x 7.8 / 4/3 p r3;
solve for r = 4.19 cm; therefore thickness = 0.19 cm.
I don’t know how I got 0.25 cm last time. Must have been dreaming!

Text box page 161
Complex Reasoning Puzzle
A 30 cm wooden ruler was placed on a bench with 10 cm overhanging. A page of the Courier Mail (58 cm x 40 cm) was placed on top of the ruler on the bench. Calculate the weight of air on the paper and predict what will happen if the overhang is given a sharp blow with your fist. You won’t believe the weight. Quick now – is it more than the weight of three Corollas?
ANSWER: F = PA = 101.3 x 103 Pa x 0.58 m x 0.40 m = 23502 N. A Corolla has a mass of about 1200 kg (12000 N) so the weight of air is more than two Corollas but fewer than three.

Text box page 161
Complex Reasoning Puzzle
A cardboard tube is placed halfway into a container of puffed wheat.
What do you think will happen when you blow air across the top of the tube. But why and what does Bernoulli have to do with it?
ANSWER: the puffed wheat gets sucked up and out the top of the tube. Bernoulli found that when air passes over a column of air you get a lowering of pressure (venturi effect). This is how a carburretor in a car works.

Text box page 167
Complex Reasoning Puzzle
Imagine you place a finger under either end of a ruler and a coin is placed on one end.
You pull away both fingers and the ruler and coin fall together staying in contact. But if you just pull away the finger under the coin something odd happens. What and why? Try it.
ANSWER: The ruler falls faster than the coin. This is because one end of the ruler stays still so the other end accelerates faster than normal (the centre of gravity of the ruler – the 50 cm mark – accelerates at the normal rate).

Text box page 168
Complex Reasoning Puzzle
There are several types of ‘crooked’ dice used by cheats. For each one described, deduce why they are crooked.
1. Green’s Load (1880) – two spots drilled out and mercury added.
2. Tapping dice – hollow centre filled with mercury but with small tube to one corner. Tap to make them crooked.
3. Bevelled – rounded on some edges.
4. Slick – one surface highly polished.
5. Hot iron – ridge along one edge.
6. Capped – one face capped with rubber.
ANSWER: 1. That face is heavier and will tend to stay face-down; 2. That corner will be heavier when tapped; 3. The bevelled edges make rolling easier so those corners will tend to roll on the surface and go face-up; 4. Polished surface will tend to roll and go face-up; 5. The edge with the ridge will not tend to roll and so it will go face-down; 6. The rubber edge will tend to bounce and go face-up. If you want to find out all the cheating methods in dice games read "Scarne on Dice" by John Skarne. It's up the 8th edition and may be at your local library.

Text box page 168
Complex Reasoning Puzzle
It’s easy to stand a pencil up on its base but impossible to stand up on its point. But why? What if you could put it in a sealed container free of air currents and arranged it so that it’s centre of mass was exactly over the point - could you do it then? Still no! But what’s the physics behind the failure?
ANSWER: One answer is that the Earth is rotating so the bottom is being dragged sideways by the surface. Another answer is that the random motion of electrons in the wood will sometimes be lopsided so that more electrons just happen to be on one side for a fraction of a second. This side will start to fall.

Text box page 170
Complex Reasoning Puzzle
Can you jump off a chair on to the floor while holding a cup full of water without spilling any. Plan how you should land to do this. Hmmm! It sounds good in theory but….
ANSWER: I would hold it above my head and land with bent knees, gradually bringing my arm down so that the deceleration could be over as long a time period as possible. Just hope the Principal doesn't walk in.

Text box page 170
Complex Reasoning Puzzle
Imagine you are standing on some bathroom scales and you bend your knees quickly. Predict what would happen to the scale reading. But why?
ANSWER: The scale reading would decrease because the top half of your body would remain motionless for a fraction of a second (inertia) as your legs moved upwards. Then you would start to fall down and the scale reading would increase a bit and then go back to normal. If this were a complex reasoning question you’d need all three parts for full marks.

Text box page 172
Complex Reasoning Puzzle
A very lightweight boat 24 m long and mass of 30 kg lies still on a quiet pond. A 90 kg man walks from bow (front) to stern. How far does the boat move relative to the pond? The answer is not 72 m.
ANSWER: 18 metres.
The best way to do this is by considering centre of gravity. In isolated systems the centre of gravity remains in the same position. The centre of gravity of the boast is at the midpoint (12 m from the end); when the man stands on the end, the centre of gravity for the whole system is 3 m from the man and 9 m from the centre. When the man walks to the other end, the centre of gravity stays in the same place and so the boat moves until a point 3 m from the other end is in the same place as the original centre of gravity. We’d do some diagrams here but they are too difficult to draw with Paintshop Pro. You do one and email it to us.

Text box page 173
Complex Reasoning Puzzle
A superball is tied to a 1.5 m string and suspended vertically from a hook. It is pulled back and allowed to strike a wooden block standing on the floor. The experiment is repeated with a lump of plasticine of the same mass as the ball. One knocks the block over, one doesn’t. Which is which and why?
ANSWER: With the superball, the rebound means there is a high change in velocity (from + to -) which means a high change in momentum. This is transferred to the wood and knocks it over. With the plasticine, it just comes to a halt (v goes from + to 0) so the change in velocity or momentum is not as high.

Text box page 174
Complex Reasoning Puzzle
A superball is placed on top of a tennis ball and they are dropped together. Wow – what a funny rebound. Predict what happens and why.
ANSWER: As the tennis ball starts to rebound upwards, the superball is still coming down so it gets a huge kick upwards (more than just bouncing off a stationary surface). The superball should shoot up to the ceiling.

Text box page 175
Complex Reasoning Puzzle
A fly crashes into the front windscreen of a train and reverses its direction. Therefore, at one instant it’s velocity is zero but as it is squashed on to the window, the window’s velocity must also be zero for a short time. How could a fly stop a speeding locomotive?
ANSWER: The fly doesn’t stop the entire locomotive – just one part. The window glass would flex backwards a tiny fraction for a short time and then flex forward to catch up to the rest of the train. And what's the last thing to go through the fly's mind as it crashes into the windscreen. That's right - its abdomen.

Text box page 175
Complex Reasoning Puzzle
An egg timer is placed on an electronic balance and it’s mass recorded when all the sand is at the bottom. It is inverted. As the sand falls, will it weigh the same or less?
ANSWER: Weighs the same. Even though some sand is in the air, other sand is crashing into the bottom and transfers momentum (adds extra weight) to the balance. These two cancel out.

Text box page 179
Complex Reasoning Puzzle
When you blow through a bent drinking straw it recoils away from the jet of air. But when you suck air in does the reverse happen? I think not! Explain that one in terms of momentum. Try it and see.
ANSWER: When you blow, the air comes out in a stream in one direction so there is a nett force in the opposite direction. When you suck, the air comes in from all directions so there is no nett force.

Text box page 186
Complex Reasoning Puzzle
If everyone faced the same way on Earth and took a step at the same time would the Earth’s rotation change? What data would you need to calculate this mathematically?
ANSWER: Radius of the Earth; force applied; duration of force. Calculate torque (T = Fr). Do calculation on change in rotational momentum of the spherical Earth. With modern methods it could be observed. We already know that the length of the day has changed by 0.0047 s in 20 years.

Text box page 187
Complex Reasoning Puzzle
A piece of wire has an eye at one end with a small wire hoop through it.
When it is rotated in a drill the hoop lifts up. Now why is this?
ANSWER: In Physics, there is a principle that when energy input continues into an object, the object tends to go to the state of maximum inertia, so the loop goes into the lifted position. Rotational inertia is the sum of the product of mass times radius squared for each atom in the object. Physicists have simplified this into simple formulas for regular-shaped objects. If you could get the formulas for rotational inertia you’d see that the horizontal loop has a higher rotational inertia. The adage "I is a maximum" is how physicists explain it. This is different if the object doesn’t have energy continually put into it. See the next question.

Textbox page 188
Complex Reasoning Puzzle
If you take a chocolate covered almond and spin it around while it is lying flat an amazing thing happens (usually) – it stands on its end. Explain this phenomena in terms of rotational inertia. Before you try it, predict if the fat end or the pointy end will be on the top. What would happen with a Smartie or M&M?
ANSWER: When left lying on a table, a chocolate covered almond lies on its side with its long axis horizontal. This is its "natural" position (with its centre of gravity low as possible) because the energy of a system tends towards a minimum. Its GPE is low as it can be. But when it is spun it stands on its end with the long axis vertical. The centre of gravity is now much higher. This seems to violate the "lowest energy" principle (see the question above). However, in cases where energy is no longer being put in (and the object is said to be dissipating energy), the physics principle "rotational inertia tends towards a minimum". Lower rotational inertia is in the upright position. Yes it happens to Smarties, M&Ms and footballs. Summary: Dissipative systems – Inertia goes minimum; conservative systems – Inertia goes maximum.

Text box page 190
Complex Reasoning Puzzle
A large ball bearing is placed…etc
ANSWER: The ball will stay in its same place relative to the desk. Friction causes the ball to rotate but it should just stay in its same place because of its rotational inertia (conservation of rotational momentum). Try it and you’ll see.

Text box page 194
Complex Reasoning Puzzle
Why do you lean forward when you get up out of a chair?
ANSWER: You need to keep your centre of gravity directly above your feet otherwise there will be a rotating force causing you to tumble backwards. Leaning forward brings your body mass over your feet.

Text box page 194
Complex Reasoning Puzzle
In 1916, a Dr Taylor observed a man carrying 40 kg ‘pigs’ of iron 11 m up a 2.4 m high incline to a train carriage. He carried 1156 pigs in 10 hours. The man’s mass was 65 kg and he rested for 15% of the time. What was his average power output for the 8 ½ hours? On a later occasion and without a rest he could only carry 305 pigs in the 10 hours. By what factor was his power output increased when he had proper rest? Suggest why cyclists use a sprint-coast-sprint sequence.
ANSWER: (a) 93 watts. P = W/t = EP/t = mgh/t = (65 + 40) x 9.8 x 2.4 x 1156 trips/30600 seconds = 93 W; (b) 1156/305 = 3.79 times

Text box page 200
Complex Reasoning Puzzle
Rod Cross, a physicist from University of Sydney, has analysed the power centres of a tennis racquet.
Dead Spot – 5 cm from top end. No bounce but maximum power for serve. All KE goes into ball.
Centre of percussion – centre of head. No ringing. Medium bounce.
Power spot – 5 cm in from bottom of head. Good for returning a fast ball but no good for serve. Maximum bounce.
Power servers like Pete Sampras want to get maximum speed for the ball so they use the dead spot for the serve for two reasons. One is mentioned above (what is it?). But what is the other one? Clamp a racquet and drop a tennis ball from 50 cm on to the three points. How does the bounce compare?
ANSWER: Maximum power; ball comes down over net at steep angle.

Text box page 204
Complex Reasoning Puzzle
A plastic measuring cylinder has two holes near the base, one twice the diameter of the other. The cylinder is filled with water and the small stopper removed. It takes t1 seconds to empty. When repeated with just the big stopper removed it takes t2 seconds. How long will it take with both stoppers removed? Do it algebraically first before you wreck a good measuring cylinder.
ANSWER: 1/5 of t1 (= 4/5 t2)
The small hole has an area of p r2;
The big hole has an area of p (2r)2 = 4p r2;
The big hole has 4 times the area so has 4 times the flow rate of the small hole;
So 1/5 of the total volume flows through the small hole in the same time 4/5 of the total volume flows through the big hole;
Hence 1/5 of t1 = 4/5 of t2 = total time.

Text box page 206
Complex Reasoning Puzzle
Evaluate the conclusion: "The Less you Know, the More you Make" etc.
ANSWER: It’s true. Bill Gates found this out when he was 20.

Text box page 209
Complex Reasoning Puzzle
NOTE: Typing error. The length of x should read "3/5 L, not 2.5 L". And point P is meant to be the pivot point not the suspension point. Sorry folks.
A pendulum bob on a string is allowed to fall but the string strikes a peg.
If the length of x is 3/5 L, the bob has zero velocity when it gets to the pivot point P. Can you prove this mathematically?
ANSWER: This pendulum is called Galileo’s pendulum after the inventor. The GPE of the ball at release is mgL. At the bottom of its travel, the bob has converted all this GPE to kinetic energy ( ½ mv2). To just return to a position directly above the pivot on a shortened string of radius "h", the bob must have critical velocity at the bottom of its travel (vcrit = Ö gr; hence:
the initial GPE (mgL) = mg2h + ½ mv2; mgL = mg2h + ½ mgh; hence h = 0.4 L (or 2/5 L).

Text box page 212
Complex Reasoning Puzzle
Per kilogram, humans…etc.
ANSWER: The Sun puts out more total heat.
Solution. Humans: Assume the average human has a mass of 50 kg. The energy output per kilogram in a day is thus 12 MJ/50kg = 2.4 x 105 J/kg. There are about 7 billion people on Earth so the total energy output is 7 x 109 x 50 x 12 x 106 J/day = 4.2 x 1018 J/day.
The Sun: has 1/10000 the output per kg of humans, hence the Sun puts out 2.4 x 105 J/10000 = 2.4 x 101 J = 24 J/kg per day. The Sun has a mass of 2 x 1030 kg (see Chapter 6) hence the total output of the Sun is 24 x 2 x 1030 = 4.8 x 1031 J/day. This is greater than all the humans put out by 1013 times (10 trillion).

Text box page 213
Complex Reasoning Puzzle
A spring has an unladen length of 10 cm. With a mass of 1 kg added it stretches to 20 cm. Three springs identical to the first one are arranged in the pattern shown as in the diagram. How far down will they stretch when 1 kg is added?
ANSWER: Down to the 35 cm mark.
The top spring will stretch 10 cm; the bottom two will stretch 5 cm.

Text box page 214
Complex Reasoning Puzzle
A spring 20 cm long has a spring constant (modulus) of 86 N m-1. If the spring is cut into two 10 cm lengths, what will the modulus of each half be?
ANSWER: the same (86 N m-1)

Text box page 215
Complex Reasoning Puzzle
A spring is 10 cm long. When a 0.6 kg mass is attached and let fall the spring stretches by 0.76 cm. Do you have enough information to calculate the spring constant?
ANSWER: Yes. The 0.6 kg mass falls through 0.66 m and hence transfers gravitational potential energy to the spring. The gravitational potential energy GPE = mgh = 0.6 x 10 x 0.66 = 3.96 J. This is transferred to elastic potential energy EPE = ½ kx2. Hence 3.96 = ½ k (0.66)2; k = 18.2 N m-1.

Text box page 224
Complex Reasoning Puzzle
During World War II, Nazi scientists threw many prisoners overboard into the freezing waters of the North Sea to see how fast their body temperatures dropped and how long it would take for them to die. Today, such data is needed by ocean rescue researchers to help develop safety devices for ocean users. Should we use this despicable ‘Nazi science’? Develop an argument for or against its use.
ANSWER: the main lines of argument usually involve: "the data has been collected so why not use it to save lives and the deaths of the prisoners would not be totally in vain" or "using it says that what was done was okay and may encourage others to do the same".

Text box page 224
Complex Reasoning Puzzle
The table below shows the effect of changes to body temperature…etc.
Death is defined as a failure to revive on rewarming above 32° C.
When people freeze to death in cold water it has been reported that they don’t seem to be in pain as they die. They often seem relaxed. What could be happening here?
ANSWER: hallucinations probably.

Text box page 225
Complex Reasoning Puzzle
At what temperature will ° C and ° F readings be the same? Could the kelvin temperature reading ever be the same as ° C or ° F reading?
ANSWER: (a) -40° C = 40° F. Use ° C = (° F –32) x 5/9; or T = (T-32)5/9; T = -40; (b) The kelvin temperature could not be the same as the Celsius as T = T – 273 has no solution; The kelvin temperature could be the same as the Fahrenheit temperature because –654K = -654° F, but this may be okay to your Maths teacher but not to a physicist (you can’t have negative kelvin temperatures).

Text box page 228
Complex Reasoning Puzzle
When you eat ice…etc.
ANSWER: Yes eating ice would help you lose weigh; you could say it has negative joules. I suppose drinking liquid nitrogen (-196° C) would do the same. You’d have to be a bit of a loser to try any of these. There’s an old story that celery has negative joules because the energy expended in chewing is more that the energy content of the food ingested. It must be true…John Laws said so.

Text box page 236
Complex Reasoning Puzzle
Imagine a 1 cm square….etc.
ANSWER: No, you can’t detect it because your body is so used to it. Just like you didn’t reaslise that the chair you were sitting on exerts pressure on your backside until we just mentioned it. If the air blows stopped you would notice it. You’d also feel yourself puffing up like a balloon as the outside pressure decreased. But you wouldn’t explode like people say as your skin is strong enough. But you’re ears and eyes would bleed. They used to kill cats and dogs this way but not anymore.

Text box page 237
Complex Reasoning Puzzle
A matchstick is placed in a test-tube of water. When you place your thumb over the top and press down, the match sinks. Propose a reason for this and test it to see if we’re lying or not.
ANSWER: The gas inside the pine match compresses and the match becomes less buoyant.

Text box page 244
Complex Reasoning Puzzle
French scientist….etc.
ANSWER: Yes, he was correct. You could have two equal masses of gases in two equal sized containers and have different pressures. For example, 32 grams of oxygen gas (1 mole) in a 22.4 L container exerts a pressure of 1 atm (at 0° C). If you put 32 grams of hydrogen gas in there, you would have 16 moles and the pressure would be 16 atm.

Text box page 247
Complex Reasoning Puzzle
When a rod made up of four metals as shown in the diagram is heated, which of the following diagrams represents its final shape. What about if it is cooled?
ANSWER: C; if cooled then B
 

Text box page 247
Complex Reasoning Puzzle
The General Electric building…etc.
ANSWER: (a) a = 3/(6000 x 61) = 8.2 x 10-6;
(b) use slots rather than holes;
(c) imagine the centre bows out so that each half is like a long skinny triangle: then the bow out length will be 94.9 mm (whew, that’s a big bow).


Text box page 249
Complex Reasoning Puzzle
The coefficient of volume expansion (b ) for iron is 0.36 x 10-4. How many times greater is this than the coefficient of linear expansion (a )? Propose a proof for why this should be.
ANSWER: three times. It is difficult to show the proof of this but we’ll try.



Let the original length of each side be L and the expansion be D L.
The increase in volume (D V) is made up by the expansion of the three faces 3(L x L x DL) + the expansion of the three side corners 3(D L2 x L) + the expansion represented by the small cube in the top front right corner DL3.
Hence: D V = 3(L x L x D L) + 3(D L2 x L) + D L3. Because D L2 and D L3 are so incredibly tiny, we can cancel them out of the equation. That leaves us with D V = 3(L x L x D L); and as b = D V/VD T, this becomes 3(L x L x D L)/L3D T or 3D L/LD T which equals 3a .

Text box page 248
Complex Reasoning Puzzle
In days gone by, warships had cannons mounted on the decks with the iron cannonballs resting in shallow brass ashtray-shaped containers called ‘brass monkeys’. These would invariably fill with water as the sea washed over the decks. In the Atlantic ocean sometimes it would get so cold that sailors would say: "It’s cold enough to freeze the balls off a brass monkey". What do you suspect they meant?
ANSWER: when the weather was cold the water on deck and in the brass monkeys would freeze and expand. This would cause the balls to roll off, hence the saying.

Text box page 253
Complex Reasoning Puzzle
Cooks sometimes put a metal skewer through potatoes to make them cook quicker. Would you speed things up by using a skewer of twice the diameter or two of the smaller skewers? If you used two, where is the optimal place to put them? Why?
ANSWER: We would say use two skewers, each about one-third of the way in from the skin. It is pretty subjective but the four sources of heat (left and right side and left and right skewer) are about equally spaced. It would make a good experiment.

Text box page 254
Complex Reasoning Puzzle
A copper rod is placed through a hole in a piece of pine and heated. Charring occurs more along the grain that across it. Now why is this? Propose a physics explanation.
ANSWER: The grain of wood is the direction the water and nutrient tubes run. When wood is heated, heated, the heat can travel more easily along these tubes than it can across the grain where the wood fibres are packed tightly together and act as an insulator. If you want more information ring the Timber Research Advisory Committee in Brisbane on 3358 1400.

Text box page 254
Complex Reasoning Puzzle
Four thermometers as shown are placed in the Sun for 10 minutes. List them in order from highest reading to lowest. Explain why.
ANSWER: Black, white, tight cotton, loose cotton. Black is a good absorber of heat, white is poor. The loose cotton would stop the thermometers warming up but the tight cotton would allow them to heat up a bit. This would make a good experiment.

Text box page 254
Complex Reasoning Puzzle
You have three ice-cubes of the same mass. Which one will melt first? Why?
ANSWER: The hollow cube (has the biggest surface area so can dissipate heat the quickest).

Text box page 254
Complex Reasoning Puzzle
How can you cook a hamburger thoroughly in the quickest time? Would you cook it on an open grill (large heat, but some charring) or in a pan (small heat). Explain using physics principles. Suggest to your Physics teacher that you have a end-of-term BBQ and the school pay for the hamburger patties. Good luck!
ANSWER: Cook it slowly so no charred meat is formed (charring acts as an insulator).

Text box page 255
Complex Reasoning Puzzle
Is it better to try to cool water fast by leaving ice float or keeping the ice submerged. Provide the physics principles behind this.
ANSWER: Allowing it to float. The cold water sinks and the warmer water at the bottom rises. If the ice was held at the bottom, the cold water wouldn’t rise (by convection).

Text box page 256
Complex Reasoning Puzzle
Imagine you added equal volumes…etc.
ANSWER: the oil would rise to the higher temperature because it’s specific heat capacity is lower (which means it takes less heat to change its temperature by 1° C than it does for an equal mass of water). They both have the same thermal energy as equal amounts of heat energy was added from the hotplate (assuming the beakers were identical).

Text box page 257
Complex Reasoning Puzzle
80 mL of cold water is placed in a polystyrene cup and the cup is placed in a beaker of hot water. Thermometers are placed in both containers. Predict the shape of the temperatures versus time graph.
The experiment is repeated but a small cube of ice is place in the cold water. Now show how the graph shapes will change.


Text box page 259
Complex Reasoning Puzzle
The temperature around a bunsen burner is lower the further you are from it. But does the temperature fall away evenly in all directions? Draw a bunsen flame as in the diagram below and predict where points of similar temperature will be, keeping in mind that both convection and radiation are operating. Join these points of equal temperature – they are called ‘isotherms’ (Gk. ‘iso = ‘equal’; thermos = ‘heat’). Try it!
ANSWER: because of convection currents the region above the flame should be much hotter than air at the same distance to the side of the flame.

Text box page 262
Complex Reasoning Puzzle
Two houses are covered in snow on their roofs. In which house is the ceiling insulation better: the one in which the snow melts quickly or slowly?
ANSWER: the one with the least snow because heat from inside the house is not getting through the roof to melt the snow.

Text box page 262
Complex Reasoning Puzzle
Cut a grape almost in half…etc.
ANSWER: Grape juice is an electrolyte and can conduct electricity from one half to the other via the skin bridge. When the microwave oven is turned on, the waves heat each half and also make charge oscillate across the skin bridge from one half to the other (that’s just what microwaves do). This makes the bridge very hot and it soon catches on fire so you now have electric current passing through flames. This causes the gas to ionize and the gas itself begins to conduct electricity which causes a brilliant arc light lightning. Cool! For further explanations of microwave oven physics see the "How things work" webpage by Louis Bloomfield, Professor of Physics, University of Virginia.

Text box page 321
Complex Reasoning Puzzle
You are listening to a radio broadcast of a live orchestral concert in London 20 000 km away. Would you hear it before or after a person at the rear of the concert hall 50 m away from the orchestra? (Sound travels at about 330 m s-1).
ANSWER: Before. It would take 20000 x 103/3 x 108 = 0.067 s to hear it by radio but would take 50/330 = 0.15 s to hear it live.

Text box page 321
Complex Reasoning Puzzle
Stealth bombers…etc.
ANSWER: Most radar absorbing materials are partially conducting plastic composites. As a radar microwave penetrates these composites, the electric field in the wave drives charges back and forth through the composites. Since they don’t conduct electricity well, they turn the wave’s energy into thermal energy and thereby absorb it. Because there will also be some reflection of the microwaves, the aircraft are designed to deflect the reflected wave away from the radar transmitter so that it’s receiver won’t detect the return wave. Stealth bombers rely on rectangular-based pyramid shaped cladding which tends to present a very low profile to radar. We’ve got no idea of how to improve the detection of a Stealth bomber otherwise we’d be millionaires and not Physics teachers.

Text box page 322
Complex Reasoning Puzzle
You can hear popcorn popping from the outside of a microwave oven…etc.
ANSWER: Microwaves are reflected off the metal grid on the door of the oven. This reflection occurs because the wave’s electric field pushes charges inside the metal walls and causes those charges to accelerate. These accelerating charges redirect (absorb and reemit) the wave in a new direction – a mirror reflection. The waves actually enter the holes of the metal grid on the front and penetrates a short distance before it dwindles to insignificance. By the time the wave has travelled a millimetre through the hole its power is halved. The rule is: if the electromagnetic wave’s wavelength is significantly larger than the hole, it won’t pass through. Sound, on the other hand passes through the walls because it causes the walls to vibrate and they transmit the sound to the outside.

Text box page 323
Complex Reasoning Puzzle
A microwave oven doesn’t heat evenly…etc.
ANSWER: 12.2 cm (l = v/f = 3 x 108/2.45 x 109 = 0.122 m = 12.2 cm).
Microwave ovens are about 36 cm wide which equals three wavelengths. A diagram should have a standing wave 36 cm long with 5 nodes in between and about 6 cm apart. Hence, the hotspots will be every 3 cm across the carousel. Try it for yourself. Get some 35 mm film containers, put equal amounts of water in each and arrange them across the insides of the oven. Take their starting temperature. Turn it on for 10 seconds and check the new temperatures. Try it without the carousel. Careful if you heat too long. The water can become superheated (over 100° C) and boil explosively when you move it.

Text box page 331
Complex Reasoning Puzzle
In the Song of the White Horse by David Belford, the lead soprano is required to breathe in helium to reach the extremely high top note. Question: if you released some of the helium into the middle of the orchestra, what would happen to the pitch of the following instruments: wind, brass, strings, percussion?
ANSWER: The speed of sound in helium is 965 m s-1 so the frequencies of the standing waves in pipes would increase (f = v/2L). No change in pitch would occur for the others. You could create have in a orchestra by releasing helium during a performance. We’re only joking of course.

Text box page 337
Complex Reasoning Puzzle
If you told a violinist that you’re a physicist and they should play the strings about one-seventh of their length from the end what would they say? Measure where they play – is it 1/7 th?
ANSWER: We’re not sure what they’d say – they’d probably tell you to get lost - but this is where a good violinist would play. If they don’t, maybe they’re not very good (but don’t say that either). What's the difference between a violin and a viola? A: violas burn longer.

Text box page 338
Complex Reasoning Puzzle
What is the squeaky sound when you are washing your hair with shampoo and what is the similarity with the shrill sound beginners produce on a violin?
ANSWER: Longitudinal waves are produced instead of transverse ones. The stiffness of the hair strand (the "string") longitudinally is very high so the frequencies are also high. That's why teenage boys don't wash their hair very often.

Text box page 338
Complex Reasoning Puzzle
You’ve dipped your finger in some wine (or metho) and run it around the rim of a wineglass. A loud sound is produced. Why doesn’t it work if your finger is not degreased with the wine?
ANSWER: Your finger has to pull the glass in a longitudinal direction so it has to stick a bit. If it is not degreased, then it will slip.

Text box page 340
Complex Reasoning Puzzle
An orchestra tunes up at the start of a concert but as the theatre warms up they have to retune their instruments. Do they find that the pitch of their instruments rises or falls as the theatre warms up. String musicians can change the tension of the strings. What do wind musicians do?
ANSWER: Wind imusicians tune by adjusting the length of the pipe. E.g.. Clarinettists and flautists can pull out at the barrel when sharp (which is usually the case). In cases of flatness, faster air generally cures the problem, although adjusting the angle of blowing can also fix the tuning in flutes (i.e.. the more it turns in, the flatter it is). Oboists and bassoonists are able to adjust the way the double reed fits into the instrument. (Answer by Ruimin Gao - gao@myplace.net.au).
 

Text box page 342
Complex Reasoning Puzzle
If you hold the right tuning fork up to your mouth cavity you can cause the cavity to resonate. Would you expect boys or girls to have the lower resonant frequency? Why?
ANSWER: Boys. Big cavity (in their mouth) therefore lower fundamental. People without brains should have a different sound when tapped on the head compared to normal people.

Text box page 344
Complex Reasoning Puzzle
When the physics laboratory is quiet, a dropped pin can be heard clearly at the back of the room. Calculate the energy arriving to the ear of the person at the back. If you want second-hand data, assume the pin has a mass of 0.2 gram and it is dropped from a height of 1.0 m (use GPE = mgh). Assume all GPE is tranformed into sound energy that radiates outwards as a large spherical surface (Asphere = 4p r2). Calculate the amount of energy per square centimetre at the back of the room (say r = 5 m). Is it more or less than 10-9 J/cm2? Not much huh?
ANSWER: Less
GPE = mgh = 0.2/1000 x 10 x 1 = 2 x 10-3 J
V = 4p r2 = 4p 5002 = 3.14 x 106 cm2;
Energy/cm2 = 2 x 10-3 J/3.14 x 106 cm2 = 6 x 10-10 J

Text box page 345
Complex Reasoning Puzzle
Tacoma Narrows bridge
ANSWER: you should get various motions like standing waves on a string. In the "Urban Legends" webpage (www.urbanlegends.com/science/bridge_resonance.html) they argue that the motions are very complex. Too complex for me I'm afraid.

Text box page 348
Complex Reasoning Puzzle
The frequency shift effect was first proposed by Doppler in 1842 but the first experiment wasn’t done until French scientist Buys-Ballot had a go in 1845. He had a carriage full of brass musicians go past him in a train as they blew a steady note. To study this effect quantitatively, what sort of measuring devices would he need?
ANSWER: Something to measure speed (tape measure, watch); something to measure frequency (nothing invented then – he would have to use a tuning fork or string and compare by ear).

Text box page 357
Complex Reasoning Puzzle
You walk towards a plane mirror at 1 m s-1. How fast does you image approach you? The mirror now approaches you at 1 m s-1. How fast does your image approach you now?
ANSWER: (a) 2 m/s; (b) 2 m/s.

Text box page 359
Complex Reasoning Puzzle
In the April 1984 edition of New Scientist magazine, a report appeared on the work of British inventor Charles deSelby… etc.
Was deSelby a big liar or what? What is wrong with his theory?
ANSWER: this is an April Fool’s joke in New Scientist.

Text box page 382
Complex Reasoning Puzzle
In 1621, French …etc.
ANSWER: The best we can do is the following diagram. If the 12 o’clock position is 0° , then the rays to the right are at 78° and 83° ; the bottom rays are at about 185° and 225° but there is a lot of room for error if you use a small protractor.


Text box page 385
Complex Reasoning Puzzle
When you squeeze…etc.
ANSWER: The bubbles look like silver balls as light is totally reflected off the surface of the bubble. They look great.

Text box page 388
Complex Reasoning Puzzle
Have you ever taken a photo…etc.
ANSWER: It should be very hard to tell the difference. We’ll add some photos here one day to show you.

Text box page 392
Complex Reasoning Puzzle
You’ve made an air lens by taping two watchglasses together. This does nothing in air but what will it do underwater? "Myopic people see better underwater" – account for this.
ANSWER: Act as a diverging lens. Myopic people need diverging lenses to spread the rays out a bit so they don’t focus in front of the retina. In water, the difference in refractive index between water and the cornea is less than for air and the cornea so the rays don’t bend as much. The rays will focus on the retina and not in front as they do in air.


Text box page 394
Complex Reasoning Puzzle
A concave mirror and a concave lens….etc.
ANSWER: The concave lens has a longer focal length as the difference in refractive indices is no longer as great so the deviation is less. The mirror is unchanged.

Text box page 396
Complex Reasoning Puzzle
Plot a graph of v (y-axis) against M (= v/u) on the x-axis using suitable data. Prove that the intercept on the y-axis equals the focal length. But what does the slope of the line equal? You’ll be surprised and delighted.
ANSWER: Proof:
1/f = 1/u +1/v
1 = f (1/u +1/v)
v = f (v/u + v/v)
v = f (v/u + 1)
v = f v/u + f, which is in the form:
y = m x + c, hence: the slope (m) = f, and the intercept (c) also = f.


Text box page 402
The word pupil comes from the Latin pupilla = ‘a doll’. When you look at yourself in someone’s eye you see a small doll-like image of yourself. Now someone was really creative with language. Quick now, is the image upright or inverted?
ANSWER: upright images are formed in a convex mirror.

Text box page 403
Complex Reasoning Puzzle
Two eyes are better than one for depth perception. Would three be better than two. Why or why not?
ANSWER: The brain relies on triangulation to estimate distance. Only two rays are needed. Three eyes would make no difference.

Text box page 403
Complex Reasoning Puzzle
You blink on average once every 5 seconds while you are awake. How many megablinks per annum is this? When you go into a shopping centre your blink rate falls to one every 12 seconds. Propose a reason for this.
ANSWER: 6.3 megablinks/annum. The atmosphere (lighting, music, colours) is conducive to making you feel relaxed so your blink rate slows down.

Text box page 403
Complex Reasoning Puzzle
There are three forms of colour blindness: protanopia, deuteranopia and total. Find out what they mean then propose a reason for the names based on the meaning of the prefixes: pro = one, deut = 2.
ANSWER: Both refer to a defect called "red-green colour blindness". It is a genetic disorder and is sex-linked so it only afflicts males (about 1 in 10). Protanopia means you have a defective perception of red. It is sometimes called red blindness. Deuteranopia means you have a defective perception of green. ‘Pro’ means one and refers to red as red is the first of the primary colours. ‘Deut’ refers to green as green is the second of the primary colours.

Text box page 405
Complex Reasoning Puzzle
In 1970 Dr Fyodorov of the Soviet Union treated a near-sighted man who had glass slivers in his eye. After the operation his near-sightedness had been cured. Prospose what the ‘radial keratomy’ operation did.
ANSWER: Near (or short) sight results in a focal point in front of the retina. This can be corrected by spreading the rays out a bit using a diverging (convex) lens. The radial keratomy would have removed some of the surface of the lens making it less curved and thus increasing the focal length.

Text box page 418
Complex Reasoning Puzzle
In the mid-1700s, French experimentalist François du Fay observed that a charged gold leaf was attracted by some electrified substances and repelled by others. He called the two types vitreous and resinous. Use Table 21.2 and you knowledge of what substances are classified as vitreous and resinous to decide if the resinous rod would have a positive or negative charge.
ANSWER: Glass is vitreous so it was the positive charge. Amber is resinous so it is negative.

Text box page 419
Complex Reasoning Puzzle
In 500 BC, Greek philosopher Thales of Miletus noticed that cork dusk was attracted to a charged amber rod. It wasn’t until 1500 (almost 200 years later) that people noticed that after a while the cork dust is repelled. Why is the dust repelled and why do you suspect that people didn’t notice it earlier?
ANSWER: Greeks didn’t do experiments. Philosophers such as Plato and Aristotle said the majority of men laboured "in order that the minority, the elite, might engage in pure exercises of the mind – art, philosophy and politics". Experiments were not pure exercises of the mind so they didn’t do them. They believed that manual labour was for the slaves. Possibly also, they may have noticed how cork dust was repelled but as it made no sense in terms of their theories of matter, they politely ignored it.

Text box page 423
Complex Reasoning Puzzle
When sand falls through a plastic funnel on to a metal plate on a electroscope, the leaves diverge. Explain this if you can.
ANSWER: The sand acquires a charge as a result of electrification by rubbing against the plastic. According to the triboelectric series (page 418) glass, and hence sand, has a lower affinity for electrons than polythene so the sand acquires a positive charge. Try it and see. Let us know if it doesn’t become positive.

Text box page 426
Complex Reasoning Puzzle
In his book, The Ascent of Science, Professor Brian Silver…etc.
ANSWER: 1.3 x 10-3 N (not 1000 N).
Solution: moles of sugar molecules = m/M = 5/342 = 0.0146 mole;
1 molecule of sugar has 270 electrons (72 from C6, 22 from H22 and 176 from O11);
Hence there are 0.0146 x 270 = 3.947 moles of electrons;
Which equals 3.947 x 6 x 1023 = 2.368 x 1024 electrons;
In this there are 2.368 x 1024/109 = 2.368 x 1015 billion electrons;
If this many are taken from one cube to the other, the nett charge on each cube will be 2.368 x 1015 x 1.6 x 10-19C = 3.78 x 10-4C.
The force at 1000 m = kQ1Q2/d2 = 9 x 109 x (3.78 x 10-4)2 / 10002 = 1.3 x 10-3N.
Note: if the distance was 1 metre instead of 1 km, the force would be 1300N which is about right.

Text box page 428
Complex Reasoning Puzzle
Here’s a good idea we overheard at a UFO conference…etc.
ANSWER: F = 9 x 109 (6 x 1023 x 1.6 x 10-19)2/12 = 8.3 x 1019 N (Whew, that’s big!).
Acceleration = F/m = 8.3 x 1019/(100 x 103) = 8.3 x 1014 ms-2. That’s big too – in one millionth of a second, the rocket would be at the speed of light.
The Flaw is that you couldn’t have a mole of charge in a wheelbarrow – it would fly apart. The only reason you can have a mole of hydrogen ions together is that there is a mole of negative charge mixed in to neutralise it.

Text box page 458
Complex Reasoning Puzzle
A writer in New Scientist magazine (July 1998) described how in 1938 he stayed in a country house in England and helped wind up a 1 tonne steel ball suspended on a chain into the roof space. During the evening, the ball was allowed to fall slowly, turning a generator to keep the light glowing all night. He said that this was impossible as there wasn’t enough gravitational potential energy in the ball to do this. Verify his claim by working out how long a 60 W lamp would glow of the ball was raised 5.0 m. Assume the energy conversion was 100% efficient (unlikely!). In actual fact it turns out that the steel ball didn’t turn a generator but turned an enclosed 44 gallon drum partly submerged in petrol. As the drum turned petrol evaporated and was burnt in a gas lantern.
ANSWER: GPE = mgh = 1000 x 10 x 5 = 5 x 104 J
Time (t) = W/P = 5 x 104 / 60 = 833 seconds (13.9 minutes). Not long – hardly enough time for a sherry.

Text box page 459
Complex Reasoning Puzzle
One that most people get wrong: 100W bulbs glow….etc.
ANSWER: The 40W bulb is brighter in series.
Solution: Firstly, calculate the resistance of each – that is the only thing that doesn’t change (the current and voltage will differ when in series to when in parallel).
P = VI = V2/R; Hence R = V2/P; R100watt = 2402/100 = 576 ohm; R40watt = 2402/40 = 1440 ohm;
In series, the total resistance is 576 + 1440 = 2016 ohm;
The current (I) will be V/R = 240/2016 = 0.119 A;
The power consumption and hence the brightness of each will be P = VI = I2R;
P100watt = 0.1192 x 576 = 8.2 watt;
P40watt = 0.1192 x 1440 = 20.4 watt (and therefore brighter).

Text box page 459
Complex Reasoning Puzzle
In an experiment to measure the efficiency of a microwave oven, 1 litre (1000 g) of water was placed in a icecream bucket and its temperature measured . It was then cooked on ‘high’ for 2 minutes and it’s temperature measured again. Use the formulas Q = mcD T and P = W/t to prove that the power output of the microwave (P) = 34.8 x D T. Hint: let Q = W. If a 750 watt microwave raised the temperature of 1 L of water by 20° C, calculate the efficiency of the oven.
ANSWER: Part A: Q = W, therefore: mcD T = Pt; 1000 x 4.18 x D T = P x 120;
hence P = 1000 x 4.18/120 x D T = 34.8 D T
ANSWER: Part B: For the water: Q = mcD T = 1000 x 4.18 x 20 = 83600 J
For the microwave oven: W = Pt = 750 x 120 = 90000 J
% efficiency = 83600/90000 x 100 = 92.9 %

Text box page 462
Complex Reasoning Puzzle
Have you seen the new compact fluorescent ….etc.
ANSWER: Fluorescent is cheaper in the long run.
Solution: Consider usage over 8000 hours:
(a) Fluoro uses 5.76 x 1018J = 160 kilowatt hours = $16. To this add the initial cost ($20) = $36.
(b) Incandescent uses 2.16 x 109 J = 600 kWh = $60. To this add 7 bulbs at 75 cents each ($5.25) = $65.25.

Text box page 461
Complex Reasoning Puzzle
A 240 V jug rated at 2000 W takes 1 min 45 s to heat 2 cups (500 g) of water from 23° C to boiling (100° C). Calculate its percentage efficiency.
If one litre of water was being boiled from the same temperature, propose whether the time would be exactly twice or more or less than twice. Hmmm - think of the energy losses.
ANSWER: 76.6%
Q (water) = mcD T = 500 x 4.18 x 77 = 160930 J
W (jug element) = Pt = 2000 x 105 = 210000 J
% eff = W(out)/W(in) x 100% = 160930/210000 x 100 = 76.6%
ANSWER: If 1 litre was being boiled it should take less because you would still have the same loss of heat in heating up the jug element. We tried it and got 105 seconds for 2 cups and 202 s for 4 cups. We'd love to test a lot of different volumes and plot a graph but we're afraid it mightn't be linear and this would spoil a good story.


Text box page 480
Complex Reasoning Puzzle
It is sometime said that the discharge of a capacitor is mathematically like the emptying of a bucket of water with a hold in the bottom. Explain why the two discharge curves are the same.
ANSWER: as the bucket empties the rate slows because there is a diminishing head of water (less pressure). For a capacitor, as the charge flows out, the repulsive forces get weaker and weaker so the rate also slows.

Text box page 510
Complex Reasoning Puzzle
If you heat a iron bar attached to a magnet as shown, at a particular temperature (Curie temperature) the bar falls off. Why might this be?
ANSWER: the heat disrupts the magnetic domains and the magnet is no longer a magnet.

Text box page 512
Complex Reasoning Puzzle
Four ring magnets are placed on a wooden pole as shown. If the distance between the top two is 10cm, calculate the other spacings?
ANSWER: Second spacing = 7.1 cm; first spacing = 5.8 cm.
Solution: let the magnets be numbered 1, 2, 3, 4 starting from the top. The force between the magnets can then be labelled: F12, F23, F34. Because magnet 2 has to support the weight of 1 as well as 2, the force F23 must be twice F12. As the magnetic force is in an inverse square relationship with distance: F12 µ 1/102; F23 µ 1/d2; but F23 = 2 F12, hence 1/d2 = 2 ´ 1/102; 2d2 = 102; d = 7.1 cm.
Similarly for the bottom pair; magnet 3 has to support the weight of two other magnets; F34 = 3 F12, hence 1/d2 = 3 x 1/102; 3d2 = 102; d = 5.8 cm.

Text box page 518
Complex Reasoning Puzzle
Which would reach the higher temperature…etc.
ANSWER: The magnet would because energy was needed to get the domains to align so it contains stored (potential) energy. This would be release on dissolving but the temperature change would be microscopic.

Text box page 519
Complex Reasoning Puzzle
Which would reach the higher temperature when dissolved in acid: a steel spring…etc.
ANSWER: The compressed spring – because it has stored elastic potential energy which is released on dissolution.

Text box page 519
Complex Reasoning Puzzle
We read on the Internet that magnets are fed to cows…etc.
ANSWER: We’re not sure about how true this stuff is about cow magnets but if you want to, check out the University of Minnesota’s web page at www.extension.umn.edu/Documents/J/O/JO1072.html. If this has disappeared, search for "cow magnets" + Minnesota.
About the time to dissolve: you check it out and let us know.

Text box page 520
Complex Reasoning Puzzle
Figure (a) shows a part of a field about…etc.
ANSWER: This is a tricky one because it very much depends on what field lines are meant to represent. When Michael Faraday started using these lines in his description of magnetic fields it was because he was poor at mathematics. But the use of lines became popular. They represent the direction an isolated north pole would travel in the field. So they don’t have a particular length – they curl around to meet the south pole or go off infinitely into space. So figure (a) shows a part of the field but when the magnetic field is turned off, the lines would still be there but would get fewer in number as the field weakened. They would not get shorter or have a gap between them and the magnet. So neither figures (b) or (c) are correct. The closest would be a figure with just one arrow. When the field is off there should be no arrows.

Text box page 602
Complex Reasoning Puzzle
A letter writer to the Courier Mail…etc.
ANSWER: Oh great! Radioactive waste all through space just waiting to be trapped into earth orbit. No thanks!

Text box page 602
Complex Reasoning Puzzle
Poet W H Auden wrote in his poem Marginalia…etc.
ANSWER: He meant that geologists and engineers would not challenge a dictator because they are either gutless or too wound up in their own work. To substantiate this claim you would have to document many geologists and engineers who stood by while tyrants were at work. Maybe W H Auden was bashed up by a geologist for his loud mouth and bad poetry.

Text box page 614
Complex Reasoning Puzzle
A Crookes’ Radiometer consists of four paddles suspended on a needle point in a low pressure glass container. One side of the paddle is painted black, the other side white. When placed in the Sun it turns around. Explain whether the black side moves away from the Sun or towards it (and why). It goes the opposite way near a block of dry ice (-44° C). Most people (even scientists) gave the wrong explanation.
ANSWER: The black side moves away from the sun. The air is hotter next to the black side so it expands and pushes the paddle away.

Text box page 637
Complex Reasoning Puzzle
Can a shadow travel faster than light…etc.
ANSWER: A shadow maybe able to travel faster than light but so what? What’s actually travelling? It’s not any object just the image of an object. It is not a violation of the special theory of relativity because you cannot transmit information using a shadow. This is one of the key points of the theory. Information can be transmitted by light, but a shadow marks the absence of light so no information can be transmitted. It’s like saying when you don’t speak the silence is travelling faster than sound. Silence doesn’t transmit information (except when your mum gives you the cold shoulder – her silence says a lot).

Text box page 647
Complex Reasoning Puzzle
Imagine you’ve slid into a parallel universe in which the year is 1840. A US Marshall is travelling by in a train at 0.75c and approaches two gun-fighters about to have a duel. Gunslinger A is closer to the Marshall. If the Marshall sees both men draw their guns at the same time, who actually drew first in (a) the Marshall’s frame of reference; (b) the gunslingers’ frame of reference.
ANSWER: (a) both the same as it says above; (b) B

Text box page 647
Complex Reasoning Puzzle
A physicist driving a very fast sports car is booked by police for travelling through a red traffic light. The physicist argues that because he was travelling fast with respect to the light, the colour of the light had its wavelength altered and appeared green to him. The judge said that he would let him off the charge of running a red light but would fine him 1 cent for every m/s he was travelling over 100 km/h. How much was he fined?
ANSWER: $840 000 ($839 999.72)
fo = 4.5 x 1014 Hz; f = 6 x 1014 Hz. Solve for v (= 0.28c = 8.4 x 107 m s-1). 100 km/h = 27.78 m s-1; hence he was 83999972 m s-1 over the speed limit.
 
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