The Doppler Shift - is this right?

[Shiina-kun]

Homework Statement

The Doppler shift was first tested in 1845 by the French scientist B. Ballot. He had a trumpet player sound an A 440 Hz while riding on a flatcar pulled by a locomotive. At the same time, a stationary trumpeter played the same note. Ballot heard 3.0 beats per second. How fast was the train moving toward him?

Homework Equations

f' = f/(1-Vs/V)

The Attempt at a Solution

Since it said that Ballot heard 3 bps, I added 3 to 440

443 = 440/(1-Vs/V)
>> 1-Vs/V = 440/443
>> Vs = (440/443 - 1)-V
>> Vs = (440/443 - 1)-343
>> Vs = 2.32 m/s

This doesn't seem right to me. I wasn't really sure if I should have added 3 to 440, but I didn't know how else to find the answer.

 

[Kurdt]

It seems ok. The beat frequency is of course given by $f_b = |f_2 - f_1|$.

 

[Moetasim]

Your attempt at a solution is on the right track, but there are a few things to consider. First, the equation for the Doppler shift is f' = f/(1-Vs/V), where f is the original frequency, f' is the perceived frequency, Vs is the velocity of the source, and V is the velocity of the observer. In this case, the source is the moving flatcar with the trumpet player, and the observer is Ballot himself. So the equation should be f' = 440/(1-Vs/V).

Second, to find the velocity of the train, we need to solve for Vs, not V. So the equation becomes Vs = V(1-f/f'). We know the original frequency (440 Hz) and the perceived frequency (443 Hz), so we can plug those in to solve for Vs.

Vs = V(1-440/443) = V(3/443)

Since we are given the beats per second (3 bps), we can use that to find the velocity of the train. Since the beats per second is a measure of the difference in frequency between the two trumpets, we can say that 3 bps = 3 Hz. So:

Vs = V(3/443) = 3 m/s

Therefore, the train was moving towards Ballot at a speed of 3 m/s.

It's important to note that the Doppler shift is dependent on the relative velocities of the source and the observer. In this case, Ballot was stationary and the train was moving towards him, which is why the perceived frequency was higher. If the train had been moving away from Ballot, the perceived frequency would have been lower.

Hope this helps clarify the concept of the Doppler shift for you!