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Measuring Relative Velocity (2)
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 electro-magnetic 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 TT (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 TR (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 Bís mirror in which case we could consider the times TT and TR 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 electro-magnetic radiation, for example micro-waves).  
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 electro-magnetic 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 fT and fR 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.   
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