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Relativity

"Relativistic" Combination of Velocities

Let us now consider two observers, A and B, who are both making measurements on the motion of a third body C.

If u is the velocity of B relative to A
and v is the velocity of C relative to A
...
what will be the velocity of C relative to B ?

Using a method similar to the "light flashes" method (see "Measuring Relative Velocities"), it can be shown that the velocity of C relative to B (ví) is given by

Detailed proof of this relation

Examples

1.

 

Suppose that u = -0∑5c and v = 0∑9c
Then, using the equation above gives

ví = 1∑4c/(1 + 0∑45c≤/c≤) = 0∑965c

Note that the Galilean transformation gives 1∑4c
2. Now consider the same situation but with u = -100ms-1 and v = 300ms-1
This time we find that both the Galilean transformation and Einstein's equation give ví = 400ms-1, in other words if v << c then Einstein's equation reduces to the simple relation ví = v - u
3.

Now we will consider the situation in which the same two observers are measuring the velocity of a light beam.

If u is still -0∑5c, then we have ví = 1∑5c/(1 + 0∑5c2/c2) = c

This example simply illustrates that the equation used to calculate ví agrees with Einsteinís initial postulate that the velocity of light is a constant.

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