On other pages in this section we have
casually referred to situations in which our old friends, A and B,
are moving at velocity v relative to each other, without ever saying
how we can know their relative velocity! 



Imagine, for a minute, that you are A, trying
to see how fast B is moving relative to you. 

Imagine also that you are both in outer
space, millions of km away from any other body. 

How would you make this velocity measurement? 



It is assumed that you
will arrive at the conclusion that you are going to have to use
light (or some other part of the electromagnetic spectrum). 



Let A send pulses of light at regular intervals towards B
as shown here. 



As can be seen in the
animation, if pulses of light are sent out with a certain time
period, T_{T} then, as each pulse has further to go to reach
B, they will be received by B with a longer time period, T_{R}
(or shorter time period, if A and B are approaching
each other) 

In this situation we
are going to be dealing with readings taken from two clocks in
relative motion so to simplify things we will define the constant, k 



and assume that A and B get along well
enough to agree to meet up to compare clock readings after the
experiment is over (so they can find out what the value of k was and
hence use it to measure their relative velocity). 



In order for us to find the relation between
k and the relative velocity, v, imagine a similar situation but in
which B reflects at
least the first pulse of light back to A. 



First, when A and B are very close
together, let them set their respective clocks to zero. 

We can imagine that A sends the "zeroth"
pulse of light at this time. 



A time T on A's clock, he/she
sends the first pulse. 

This first pulse will be received by B
(according to his/her clock) at time 



this is simply from the definition of k,
above. 



Now, for this reflected pulse, let's look at
things from B's point of view for a while. The situation is now
"reversed", B is sending a pulse to A who is moving away at
velocity, v. 



Therefore, the reflected pulse should arrive
back at A at time given by (the factor k "works both ways") 



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 



The light went from A to B and back in a
time given by 



which means that the distance between A and
B at the time (according to A) of the reflection is given by 



where c is the speed of light. 

A can now conclude that their relative
velocity is given by 



so, we have 



Rearranging this we find 



from which v can easily
be calculated (as long as B tells A the time period he/she measured
for the received pulses!) 



NB 

Rearranging the equation again gives 



from which we notice that, if we change v to
v (in other words, reverse the sense of the relative velocity) the
value of k changes to 1/k 



To get some idea of what value to expect for
k, let the relative velocity be half the velocity of light. 

This would give 





We can now consider that, instead of sending
separate pulses of light, A send a continuous beam of, for example,
radio waves. 

In this situation, k would be the ratio of
the received to transmitted time periods of the radio waves. 

As it is more common, in practice, to
measure the frequency of radio waves, we could ask A and B
to compare the frequencies of transmitted and received
waves. 

Remember that the relation between frequency
and time period is 



So, if the ratio of frequencies was used
(instead of k as defined above) we would have 





For a slightly different way of measuring relative velocity see
here. 
