Chapter 2 - Worked Solutions

to selected Complex Reasoning questions from the OUP text Senior Physics - Knowledge, Processes and Reasoning by Walding, Rapkins and Rossiter

RACE 1: Chris runs 100 m in time t; Sandy runs 90 m in time t
v(Chris) = 100/t; v (Sandy) = 90/t
RACE 2: s (Chris) = 110 m; s (Sandy) = 100 m
t₂(Chris) = s/v = 110t/100 = 1.10t; t₂(Sandy) = s/v = 100t/90 = 1.11t
Chris is still faster.

Mach 2.5 = 825 m/s; Mach 1.2 = 396 m/s
v = u + at
396 = 825 + a x 30
a = - 14.3 m s^-2

s₁ = 300 m; s₂ = 300 m; s₃= 300 m; s₄ = 300 m
(1) v₁² = u ² + 2as
= 0 + 2 x 0.95 x 300
= 23.87 m s^-1
v = u + at
23.87 = 0 + 0.95 t
t₁ = 25.1 s

(2) v₂ = s₂/t₂
t₂= s₂/ v₂ = 300/23.87 = 12.57 s;v₂ = 23.87 m s^-1

(3) t₃= 12.57 s; v₃ = 23.87 m s^-1

(4) v ² = u ² + 2as
0² = 23.87² + 2 x a x 300
a = - 0.95 m s^-2
t₄ = t₁ = 25.1 s
Answers: (a) Maximum speed = 23.87 m s^-1
(b) Time = t₁ + t₂ + t₃ + t₄ = 75.3 s
(c)

The four intervals are shown in the following diagram


At top of jump v₄ = 0
v² = u² + 2as
0² = u² + 2 x -10 x +0.76
u (take-off speed (v₄)) = 3.9 m/s

BOTTOM 15 cm:
v₂² = u² + 2 as₂
= 3.9² + 2 x -10 x 0.15
v₂ = 3.49 m/s
v₂ = u + at₂
t₂ = (v₂ - u)/a = 0.04 s

TOP 15 cm:
v₃ = u² + 2as
= 3.9² + 2 x -10 x 0.61
v₃ = 1.73 m/s
t₃ = (v₃ - u)/a = 0.22 s
t₄ = (v₄ - u)/a = 0.39 s

Answers:
(a) Bottom 15 cm: t₁ to t₂ = 0.04 s
(b) Top 15 cm: t₃ to t₄ = 0.17 s
(c) Therefore, time at bottom is about 1/4 of time at top.

First toss: s₁, t₁, u₁, a
Second toss: s₂, 2t₁, u₂, a
At top:
First toss: v = u₁ + at₁ = 0
u₁ = - at₁
Second toss: v = u₂ + at₂ = 0
u₂ = - at₂which equals -2at₁
Hence: u₁ = -at₁ = u₂/2

s₁ = u₁t₁ + ½ at₁²

s₂ = u₂t₂ + ½ at₂²
s₂ = 2u₁2t₁ + ½ a(2t₁)²
s₂ = 4u₁t₁ + 2at₁²

Ratio s₂/s₁ = (4u₁t₁ + 2at₁²)/ (u₁t₁ + ½ at₁²)
= (4u₁ + 2 at₁)/(u₁ + ½ at₁)
Replace at₁ by -u₁
Ratio s₂/s₁ = (4u₁ + -2 u₁)/(u₁ + ½ ₁)
= 2u₁/½ u₁
Therefore: s₂ is 4 times s₁ or second toss goes four times as high.

(a) s = -50m; t₂ = t₁ - 1; u₁ = 0; u₂ = ?

Race 1:

s = ut₁ + ½ at₁²
-50 = ½ x ^-10 t₁²
t₁ = 3.2 s; hence t₂ = 2.2 s
v₁ = u + at = 0 + ^-10 x 3.2 = ^-32 ms^-1

Race 2:

s = u₂t₂ + ½ at₂²
-50 = u₂ x 2.2 + ½ x ^-10 x 2.2²
u₂ = -11.7 ms^-1; v₂ = u +at = -11.7 + ^-10 x 2.2 = ^-33.7 ms^-1

(b)

 

Acceleration is more thrilling!
a (Kitty) = (v-u)/t = (628 x 1000/3600)/3.72 = 46.9 ms^-1
a (Eli) = (v-u)/t = (116 x 1000/3600)/0.04 = 805 ms^-1
 

Case 1: starting velocity = ⁺u

v₁² = u² + 2as
v₁ = Ö (u² + 2as)

Case 2: starting velocity = ^-u

v₂² = u² + 2as
v₂ = Ö (u² + 2as)

The formula is the same for both because when u is squared it doesn't matter if the value for u is + or -. Hence, the velocity is the same in both cases.
 

s₁ = ut₁ + ½ at₁²
s₁ = ^-5t₁²
s₂ = u(t₁ - 1) + ½ a(t₁ - 1)²
s₂ = ^-5 (t₁ - 1)²
s₂ = ^-5t₁² + 10t₁ - 5
(a) s₁ - s₂ = 5 - 10t₁
(b) v₁ = u₁ + at
v₁ = -10t₁
v₂ = u₂ + a(t₁ - 1)
v₂ = -10t₁ + 10
Ratio v₁/v₂ = -10t₁/(-10t₁ + 10) = t₁/(t₁ - 1)

Case 1

s = ut + ½ at²
400 = 0 + ½ a x 7.2²
a = 15.4 ms^-2

Case 2

v = u + at
305 x 1000/3600 = 0 + a x 7.2
a = 11.8 ms^-2
The rider probably reached top speed before the 400 m so in Case 2 where he reached 305 kmh^-1 at the finish line, he probably reached this after 5 seconds and kept constant speed from then on. In case 1, we are assuming acceleration is constant and that the top speed is irrelevant. The v/t graphs probably look like this:

 

At current mass: t₁ = 6.92s and s = 400 m

s = ut + ½ at₁²
400 = 0 + ½ a x 6.92²
a = 16.7 ms^-2

At heavier mass: 60 pounds - 60/7.2 = 27.3 kg
Time lost = 20/27.3 x 0.3 s = 0.22 s
t₂ = 6.92 + 0.22 s = 7.14 s
s = ut +½ at²
400 = 0 + ½ a x 7.14²
a = 15.7 ms^-2
 

(a) 30 knots = 30 x 100 feet in 1 minute = 30 x 100 x 60 ft per hr
= 180000 ft/hr = 180000/6080 nautical miles/hr = 30 naut mi/hr

(b) 30 knots - 180000 ft/hr = 180000 x 0.305 metres/hr = 180 x 0.305 km/h
= 54.9 km h^-1
 

t (s)	s (m) Earth	s (m) Moon	s (m) Mars	s (m) Sun
0.0	0.00	0.00	0.00	0.00
0.2	0.20	0.03	0.08	5.40
0.4	0.80	0.13	0.30	21.60
0.6	1.80	0.29	0.68	48.60
0.8	3.20	0.51	1.22	86.40
1.0	5.00	0.80	1.90	135.00
1.2	7.20	1.15	2.74	194.40
1.4	9.80	1.57	3.72	264.60
1.6	12.80	2.05	4.86	345.60
1.8	16.20	2.59	6.16	437.40
2.0	20.00	3.20	7.60	540.00

 

There are at least five solutions to this problem. Perhaps the easiest (?) one for Senior Physics students is one using internal triangles.

1. Drop perpendiculars from points B and D to line AC (see figure below):


2. Let the constant speed = v (miles/minute). Then: distance (miles) = time (minutes) x speed (v)
The distance AC (directly) = 30v; the distance ADC = 40v; the distance ABC = 35v.
Hence: AD = 40v -10; AB = 35v - 10
Let x be the distance from C to the perpendicular on AC from B.
Let y be the distance from C to the perpendicular on AC from D.
The two lower triangles are similar, hence the perpendiculars have the lengths x and y as shown in the figure.

3. Using Pythagorean theorem on the top two triangles and one of the two identical (similar) lower triangles:
Equation 1: (40v-10)² = x² + (30v-y) ²
Equation 2: (35v-10)² = y² + (30v-x) ²
Equation 3: 10² = x² + y²

4. Expand Equation 1 and substitute 10² - y² for x ²
1600v² - 800 v + 100 = 100 - y² + 900v² - 60 vy + y²
700v² - 800 v = -60vy
y = 13.33 - 11.67v

5. Repeat for Equation 2 substituting 10² - x² for y ²:
1225v² - 700v +100 = 100 - x² + 900v² - 60vx ²
325v - 700v = -60vx
x = 11.67 - 5.42v

6. Substitute these values for x and y into equation 3:
100 = (11.67 - 5.42v) ² + (13.33 - 11.67v) ²
100 = 136.1 - 126.4v + 29.38v² + 177.8 - 311.1v + 136.1v
165.5v ² - 437.5v + 213.9 = 0

7. Use quadratic formula to solve for v
Two solutions: v = 0.648 miles/minute and v = 1.996 miles/minute.

8. Convert to miles/hour (x 60): 38.8 mph and 119.8 mph.

9. Discard the second speed as too fast (break the speed limit?).
Answer: 38.8 mph

ALTERNATIVE SOLUTION:
Use the cos rule: the cosine of any angle of a triangle equals the sum of the squares of the adjacent sides less the square of the other side, all divided by twice the product of the adjacent sides.

Applying this to the figure (and simplifying) shows that:
Cosine of angle ABC = (7 - 3.25v)/6
Cosine of angle ACD = (8-7v)/6
When these two values are squared and added together they equal 1.
This follows from the rule that: when two angles add up to 90°, the sum of squares of their cosines = 1.
Simplifying and collecting terms gives a quadratic which produces two roots as in the solution above.

ANOTHER SOLUTION
Construct a rectangle as shown in the figure below:

1. Write square-of-the-hypotenuse equations for the three right triangles of which AC, AD and AB are the hypotenuses.
Equation 1 (for AC): (30v)² = (10 + x) ² + (y + 10)²
Equation 2: (for AD) (40v-10)² = x² + (y + 10)²
Equation 3: (for AB): (35v - 10)² = y² + (x + 10) ²

2. Subtract Equation 2 from Equation1 and find x in terms of v.
30v² - (40v - 10) ² = (x + 10) ² - x²
x = -35v² + 40v - 10

3. Subtract Equation 3 from Equation 1 and find y in terms of v.
30v² - (35v - 10) ² = (y + 10) ² - y²
y = -16.25v² + 35v - 10

4. Substitute these values of x and y into Equation 1.
(30v) ² = (-16.25v² + 35v) ² + (-35v² + 40v) ²
900v² = 16.25²v⁴ - 16.25 x 35 x 2 v³ + 35²v² + 35²v⁴ - 35 x 40 x 2 v³ + 40²v²

5. Simplify and divide by v², (allowable since v can't be equal to zero).
1489.1v² - 3937.5v + 1925 = 0

6. You should have a quadratic. Solve using the quadratic formula and you should get roots as in the methods above.
Pretty simple huh?
 

First statement: s µ t² is correct as our formula is s = ½ at² which is the same as s µ t²
Second statement: v µ s is incorrect as v² = 2as or v² µ s
 

tan 4.3° = 1000/s therefore s = 1000/tan 4.3° = 13300 m
298 km/h = 298 x 1000/3600 ms^-1 = 82.8 ms^-1
t = s/v = 13300/82.8 = 160 s
NOTE: the answer in the back of the book is wrong. It is not 44.6 s but 160 s.
 

Impact speeds:
Of a child in a car: v = 50 x 1000/3600 = 13.9 ms^-1
Falling from a building: (assume each story to be about 3 m high):
v² = u² + 2as
v = Ö (2 x 10 x 6) = 10.9 ms^-1
Hence there is a lower impact speed in falling from a building but the concrete is harder than a car dashboard. Probably about equal impact damage.
 

s = ut + ½ at²
64 = 0 + ½ x 2 x t²
t = 8.0 s
s _(t=0) = 0
s_(t=2) = ½ at² = ½ a 2² = 4 m
s_(t=4) = ½ a x 4² = 16 m
s_(t=6) = ½ a x 6² = 36 m
s_(t=8) = ½ a x 8² = 64 m
 

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