Question #2877

A train is at rest and you are driving toward it at 50 m/sec when it emits a pitch of 1500 Hz, with what frequency will you hear the pitch?

Expert's answer

The general Formula is

<img src="/cgi-bin/mimetex.cgi?v%20=%20v_0%20%5Cfrac%7Bs%20%5Cpm%20V%7D%7Bs%20%5Cpm%20v%7D" title="v = v_0 \frac{s \pm V}{s \pm v}">

where: s - is the velocity of waves in the medium, V - is the velocity of the receiver (driver) relative to the medium; positive if the receiver is moving towards the source, v- is the velocity of the source (train) relative to the medium; positive if the source is moving away from the receiver.

As a train is at rest than v=0. The initial frequency is 1500 Hz, so we get

<img src="/cgi-bin/mimetex.cgi?v%20=%20v_0%20%5Cfrac%7Bs%20%5Cpm%20V%7D%7Bs%7D" title="v = v_0 \frac{s \pm V}{s}">

In dry air at 20 °C (68 °F), the speed of sound is 343.2 metres per second, so we get

<img src="https://latex.codecogs.com/gif.latex?v%20=%20v_0%20%5Cfrac%7Bs%20%5Cpm%20V%7D%7Bs%20%5Cpm%20v%7D%20=%20v_0%20%5Cfrac%7Bs%20%5Cpm%20V%7D%7Bs%7D%20=%201500%20Hz%20%5Cfrac%7B343.2%20+%2050%7D%7B343.2%7D%20=%201500*1.15%20Hz%20=%201725%20Hz." title="v = v_0 \frac{s \pm V}{s \pm v} = v_0 \frac{s \pm V}{s} = 1500 Hz \frac{343.2 + 50}{343.2} = 1500*1.15 Hz = 1725 Hz.">

So the answer is ν=1725 Hz.

<img src="/cgi-bin/mimetex.cgi?v%20=%20v_0%20%5Cfrac%7Bs%20%5Cpm%20V%7D%7Bs%20%5Cpm%20v%7D" title="v = v_0 \frac{s \pm V}{s \pm v}">

where: s - is the velocity of waves in the medium, V - is the velocity of the receiver (driver) relative to the medium; positive if the receiver is moving towards the source, v- is the velocity of the source (train) relative to the medium; positive if the source is moving away from the receiver.

As a train is at rest than v=0. The initial frequency is 1500 Hz, so we get

<img src="/cgi-bin/mimetex.cgi?v%20=%20v_0%20%5Cfrac%7Bs%20%5Cpm%20V%7D%7Bs%7D" title="v = v_0 \frac{s \pm V}{s}">

In dry air at 20 °C (68 °F), the speed of sound is 343.2 metres per second, so we get

<img src="https://latex.codecogs.com/gif.latex?v%20=%20v_0%20%5Cfrac%7Bs%20%5Cpm%20V%7D%7Bs%20%5Cpm%20v%7D%20=%20v_0%20%5Cfrac%7Bs%20%5Cpm%20V%7D%7Bs%7D%20=%201500%20Hz%20%5Cfrac%7B343.2%20+%2050%7D%7B343.2%7D%20=%201500*1.15%20Hz%20=%201725%20Hz." title="v = v_0 \frac{s \pm V}{s \pm v} = v_0 \frac{s \pm V}{s} = 1500 Hz \frac{343.2 + 50}{343.2} = 1500*1.15 Hz = 1725 Hz.">

So the answer is ν=1725 Hz.

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