PLEASE NOTE: students will often say Diagram 1 represents 1/4 l so l = 48 cm. We say that there is no indication of end correction to explain the anomaly. However, in subsequent editions we will modify this question because if l is 76cm then 1/4l is 19 cm which is more than the question says. We'll change it to 21cm and 59cm thus allowing for a 2cm end correction. We're not sure why we're telling you this but it is cathartic.
Assume the three
microphones A, B, C are on a straight
line 1000 m apart as shown below. The gun must be a bit further away from B than
A as the sound is first heard at A, then B, then C. Draw the gun in position as
shown with A, B and C 10 cm apart. You cannot draw in the position of the gun as
that is what you are trying to find. The sound radiates out from the gun in a
spherical wavefront (or circular if drawn on paper in 2-D) as shown.
The sound takes 1.2 seconds to travel to B after being heard at A. Assume that the speed of sound in air is 330 m s-1, this corresponds to a distance of 330 x 1.2 = 396 m. If we draw a circle centered at B with a radius of 396 m (3.96 cm) then all points on that line represent the possible locations of a sound that is 396 m away from B at the same time.
The question also says that the sound at C is heard 3.7 s after it is heard at A. In this time a sound will have travelled 330 x 3.7 = 1221 m. So draw a circle centered on C with a radius of 12.21 cm (= 1221 m at a scale of 1:10000). Mark the intersection as point X.
The sound of the gun is heard at point X at the same time as point A so draw a circle centered at X with a radius of XA. Then draw a circle centered at A with the same radius. Where these lines intersect is the location of the gun G. The distance from the gun to A (GA) is measured as 9 cm ( = 900m). The angle BAG is 82°. The angle BCG is about 25°.
The question implied that there may be more than one solution. This is correct. An imaginary source of sound (in a mirror-image position on the opposite side of the line of microphones as in the diagram below) would also give rise to similar conditions at the microphones. This imaginary source would be ruled out by common sense.
On the diagram above, the circular wavefront is shown as it reaches microphone A. Three lines are drawn from the Gun to microphones A, B and C. As well, a perpendicular "y" is dropped from the Gun to the line of the microphones. It is "x" metres from A. We have three right angled triangles each sharing a common side (y). Three simultaneous equations can be proposed with the two unknowns x and y.
Equation 1 for DGCX: y2 = (1221 + s)2 - (2000 - x)2
Equation 2 for DGBX: y2 = (396 + s)2 - (1000 - x)2
Equation 3 for DGAX: y2 = s2 - x2
Equating Eq 1 and 3:
12212 + 2442s + s2 - 20002 + 4000x +x2 = s2 - x2
x = 42 m
Equating Eq 2 and 3:
3962 + 792s + s2 - 10002 + 2000x - x2 = s2 - x2
s = 958 m
The angle GAX = cos-1 (x/s) = cos-1 (42/958) = 87.5°
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