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). 

What follows is a thought experiment, in the way
it is described here, but leads to a
practical method of measuring relative velocities. 



The basic idea is shown in this animation. 



The following diagrams are all drawn from A's point of
view. 

This is reasonable because, in what follows, all the measurements
will be made by A (that is, in A's frame of reference). 

If we find that the velocity of B relative to A is v, then we know
that the velocity of A relative to B is simply v, the same
magnitude but in the opposite sense. 

(A slightly different way of measuring relative velocity, involving
measurements from both A's and B's clocks, can be seen here.) 



Procedure: 

When A and B are together, A sets his/her clock to zero. 



Some time later, A transmits a short pulse of light to B. 

Let the time at which this pulse is transmitted be T_{T} (on A's clock). 



B carries a mirror which reflects the pulse
back to A 



A must note the precise time that the light pulse is received
after reflection. 

Let this be T_{R}
(again, on A's clock). 





A now needs to decide
two things: 

1. What was the time on A's clock at the
instant the light reached B? 

2. At what distance, x, was B when the light
reached him/her? 

If A can decide both of
these things he/she can simply use v = x/t to find the magnitude of
the velocity of B relative to A. 



The time at which the
reflection of the pulse occurred must be half way between
the time of sending the pulse and the time of receiving its
reflection, that is 



The time between the light being transmitted
by A and being received by A is 



In this time, a light pulse will travel 



therefore, the distance x is 



This means that the relative velocity is 



From which the velocity
can be found. 

A could track the relative velocity by
sending a series of such pulses at regular intervals (as
shown in the animation) to be
reflected by B’s mirror in which case we could consider the times T_{T}
and T_{R} as being the times between consecutive pulses or
the time period. 

Obviously, the same result would still
apply. 

Now let's go one step further. 



A Practical Method
of Velocity Measurement 

Let A send a continuous beam of
light (or other electromagnetic radiation, for example
microwaves). 

We can consider the crest of each wave
in the same way as the separate pulses, so now the time periods
involved in our calculations will be the time periods of the waves. 

The equation
above can be rearranged to give the ratio of the received to
transmitted time periods 



and as it is fairly easy to measure the
frequency of an electromagnetic wave, we could put our result in
terms of frequencies, remembering that 



This means that the relation between the
received and transmitted frequencies is 



Notice that, even for
an object moving the quite high speed (by terrestrial standards) of
300ms^{1} the equation
above predicts a very small change in frequency. For this speed we
find 



This might appear hard
to detect but, in fact, quite simple electronic circuits exist which
can mix two signals together to find their difference. 

The difference between f_{T} and f_{R}
is, from the above equation, given by 



Looking at this expression, we see that, if
v << c (usually the case) we have 



where Δf/f
represents the change in frequency as a fraction of the transmitted
frequency. 
