Question

A rocket is 120 meter directly over a person and it is traveling at 500 m/s. Assume the temperature is 20 C. a) How many seconds will it take for the person to hear the sonic boom, how far has the rocket traveled in that time, how far has the sound wave that reached the person traveled, and what is the angle of the shock wave? b) Now, assume that the rocket is accelerating at a rate of 12 m/s2 . Answer the same questions and the initial and final angle of the shock wave?

I need help with the above question. I've tried drawing an illustration for it but I keep getting the wrong answer. The answers are as follows:

a) .25455 seconds, drocket = 127.2772 m, dsound = 87.31218 m, θ = 43.3143 degrees

b) .25455 seconds, drocket = 127.66377 m, dsound = 87.31218 m, θi = 43.3143 degrees , θf = 42.987 degrees

Answer 1

First of all, we will find the angle of sonic boom (shock wave). we hear the sonic boom when shock wave passes over head.

This is calculated as

sin = speed of sound / speed of airplane

speed of sound at given temperature of 20 degree is calculated using the formula

331 + 0.6 * T

331 + 0.6 * 20 = 343 m/s

sin = 343 / 500

= sin^-1 (343 / 500)

= 43.3143 degree

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Now, we have the angle of shock wave , so we can find time for shock wave to reach the person

If theta is the angle between distance covered by rocket  and the sonic boom then

tan = h / d

where h is height above the head and d is velocity * time

so,

tan = h / velocity * time

time = h / velocity * tan

time = 120 / 500 * tan 43.3143

time = 0.25455 seconds

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Now we have time so we can find both the distances ( d rocket and d sound)

for

d (rocket) = speed of rocket * time = 500 * 0.25455 = 127.2772 m

d (sound) = speed of sound * time = 343 * 0.25455 = 87.3121 m