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Measuring Relative Velocity (1)
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).  
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, TT then, as each pulse has further to go to reach B, they will be received by B with a longer time period, TR (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.  
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