two guns were fired from the same place at an interval of 10 minutes and 30 seconds but a person in the train approaching the place hears the second shot 10

Question

two guns were fired from the same place at an interval of 10 minutes and 30 seconds but a person in the train approaching the place hears the second shot 10 minutes after the first. The speed of the train in kmhr supposing that speed travels at 330 meters per second is

 

please explain me in details and concept of this question

Answer 1

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Answer 2

Hi Amey,

 

This problem deals with what is called the Doppler effect. The Doppler effect states that the frequency that you perceive from an object emitting a sound changes if the source and/or the detector (maybe you) moves. Let f be the true frequency of the sound and f' be the perceived frequency. Let v be the speed of sound, V_D be the speed of the detector and V_S the speed of the source. The most general formula for the Doppler effect is

 

f' = f(v +/- V_D)/(v -/+ V_S)

 

The rule of thumb is that if the source or detector moves toward each other, this works to increase the perceived frequency. So, if the detector moves toward the source, then we would choose the + sign in the numerator because that serves to increase f'. On the other hand, if the source moves toward the detector, we would choose the - sign in the denominator because that serves to also increase f'.

 

In this problem, the source stands still. We know this because the gun is being fired from the same location. So, we set V_S = 0, because the source does not move. You are sitting on a train approaching the gun. Since you are moving toward the source, we would choose the + sign in the numerator. We are told to use v = 330 m/s for the speed of sound.

 

We need to work out the frequencies. The true frequency is 1 shot every 10.5 min = 630 sec. So, f = 1/630. But, the person on the train perceives a frequency of one shot per 10 min = 600 sec. So, f' = 1/600. We need to solve for the detector speed, V_D, which is the speed of the train. So, with all of this, we write:

 

f' = f(v + V_D)/(v +/- 0)

 

or

 

f' = f(v + V_D)/v

 

Cross multiply

 

f(v + V_D) = f'v

 

v + V_D = f'v/f

 

or

 

V_D = f'v/f - v

 

or

 

V_D = v(f'/f - 1)

 

Plugging in numbers, we get

 

V_D = (330 m/s)((1/600 s)/(1/630 s) - 1)

 

V_D = (330)(630/600 - 1)

 

V_D = 16.5 m/s