Wednesday, 24 September 2014

Relativity: Relative Velocity


Consider two observers, A and B, moving as shown below.
.....

3ms-12ms-1
To find the velocity of B relative to A, imagine yourself to be A. You will see B coming towards you (that is, moving in the negative sense) at 5ms-1.
We therefore say that the velocity of B relative to A is -5ms-1.
Similarly, the velocity of A relative to B is +5ms-1.
It must be remembered that the velocities of the observers stated in the example above are also relativevelocities. They are the velocities of the observers relative to the earth.
Consider the following situation
.....

3ms-13ms-1
Now it is clear that the relative velocity of the two observers is 0ms-1.
By imagining yourself to be one of the observers you are giving yourself zero velocity relative to that observer. In other words, mathematically, you are removing (subtracting) the velocity of that observer from your measurement. So, mathematically
VB relative to A = VB - VA
Therefore, using the figures of the first situation, we have
VB relative to A = -2 - (+3) = -5ms-1
One of Einstein's aims in developing the theory of relativity was to be able to take measurements made by one observer and transform them into measurements made by another observer. In other words, if one observer says that a body is moving at vms-1, what will another observer say about the same body?
We will now add a third body, p, as shown below.
.......

Now you see why we chose the letter p …. yes it's a pelican.
Suppose that we do not know how fast A and B are moving over the surface of the earth but we do know theirrelative velocity. Let the magnitude of their relative velocity be u = 4ms-1.
Let v be the velocity of p relative to A has been measured.
If A tells B that p moves at v = 10ms-1 relative to him, B will conclude that the velocity of p relative to himself is v’ = 6ms-1 in the negative sense.
It is clear that again the result was found by a simple subtraction, as follows

velocity of p
 relative to 
Bv’
=
velocity of p
relative to 
Av
-
velocity of B relative to Au
velocity of p
relative to 
Bv’
=
-10 - (-4) = -6ms-1