Longitudinal Doppler Effect

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Main Question or Discussion Point

A and B leave from a common point and travel in opposite directions with
relative speed v. When B’s clock shows that a time T has elapsed, he (B)
sends out a light signal. When A receives the signal, what time does his (A’s)
clock show? Answer this question by doing the calculation entirely in (a) A’s
frame, and then (b) B’s frame.

(Y = gamma)

a)

In A's frame, when A's clock reads YT, B's clock reads T. This means B is at a distance YTv from A. When B emits the photon, the photon takes time YTv/c to reach A in A's frame. Thus, the total time for A is YT(1 + v/c).

b)

In B's frame, when his clock reads T, A is at a distance Tv away. Then, B emits a photon which travels at a speed of (c-v) relative to A in B's reference frame. Thus, the time taken for the photon to reach A is Tv/(c-v). Thus the total time this takes according to B is T + Tv/(c-v) = T(1 + v/(c-v)). By time dilation, in A the total time is YT(1 + v/(c-v)).

I get these two different answers. Does anyone know what I'm doing wrong?
 

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stevendaryl
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A and B leave from a common point and travel in opposite directions with
relative speed v. When B’s clock shows that a time T has elapsed, he (B)
sends out a light signal. When A receives the signal, what time does his (A’s)
clock show? Answer this question by doing the calculation entirely in (a) A’s
frame, and then (b) B’s frame.

(Y = gamma)

a)

In A's frame, when A's clock reads YT, B's clock reads T. This means B is at a distance YTv from A. When B emits the photon, the photon takes time YTv/c to reach A in A's frame. Thus, the total time for A is YT(1 + v/c).

b)

In B's frame, when his clock reads T, A is at a distance Tv away. Then, B emits a photon which travels at a speed of (c-v) relative to A in B's reference frame. Thus, the time taken for the photon to reach A is Tv/(c-v). Thus the total time this takes according to B is T + Tv/(c-v) = T(1 + v/(c-v)). By time dilation, in A the total time is YT(1 + v/(c-v)).

I get these two different answers. Does anyone know what I'm doing wrong?
Your calculation in B's frame is wrong. You got it correct, that the time, according to B's frame, for the photon to reach A is T_arrive = T(1+v/(c-v)) = T/(1 - v/c). But in B's frame, A's clock is running slower, so the elapsed time on A's clock is T'_arrive = T_arrive/Y = T/(Y (1-v/c)).

That's the same as your calculation in A's frame, since

T/(Y (1-v/c)) = YT (1+v/c)
 
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