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Measuring Relative Velocity

How can we measure the relative velocity of two bodies when the distance between them is very large? Whatever method we choose, it will involve the use of light or some other electro-magnetic radiation. Consider two observers, A and B, moving with relative velocity v as shown below. Notice that the diagram is drawn from Aís point of view.

A sends pulses of light to B at intervals of T seconds (as measured on A's clock). If A and B are moving away from each other, as shown in the diagram, B receives the pulses at longer intervals because each pulse has further to go than the preceding pulse. To help in measuring the relative velocity, we will define the constant, k as follows

B has a mirror which reflects the pulses back to A. We will consider the following sequence of events


At the instant when A and B are together, they set their clocks to zero and the first pulse of light is sent. In other words, the first pulse is sent and received at t = 0.


The second pulse, sent by A at time T is received by B at time kT.


The reflected pulse is therefore received by A at time k2T.

Now, according to A, the time at which the reflection of the second pulse occurred must be half way between the time of sending the pulse and the time of receiving its reflection, that is

(T + k2T)/2


This means that the light went from A to B and back in a time given by

k2T - T

Therefore the reflection occurred at the instant when the distance between A and B was

(k2T - T)c/2

So, if the velocity of B relative to A is v, we have

and this gives us a method of measuring relative velocities.

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