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Relativistic velocity addition

QUESTION: Two spaceships are observed from Earth to be approaching each other along a straight line. Ship A moves at 0.40c relative to the Earth observer, ship B moves at 0.50c relative to the SAME observer.
What speed does the captain of ship A report for the ship of ship B?
vao = speed of ship A to observer = 0.40c vbo = speed of ship B to observer = 0.50c

c = speed of light = 3 x 108 m/s

Possible choices:
(a) 0.10c
(b) 0.75c
(c) 0.85c
(d) 0.95c
(e) 0.99c


The first thing to point out is that the observer on the Earth sees the two spaceships approach each other at 0.9c. In fact if their velocities had been 0.75c and 0.85c the observer on the Earth would have seen them approach each other at 1.6c. This does not violate the principle of relativity as neither of them move with a velocity greater than c relative to the Earth.

Speed = [vao + vbo]/(1 + vao x vbo/c2) Speed = [0.40c + 0.50c]/(1 + ( 0.40c x 0.50c)/c2)

Relative approach speed = 0.75c
© Keith Gibbs 2011