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Lets suppose a spaceship travels with v = 0.9c relative to the Earth. The time inside the spaceship would pass slower than on Earth. Would the astronauts measure a different speed (that means, a different one that the observer on Earth does) in relation to the same reference frame (Earth)?

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Speed of what? What are these astronauts measuring? –  Chris White Jan 12 '13 at 1:32
    
@Chris White: I think OP is essentially asking Is the velocity of $A$ relative to $B$ the opposite of the velocity of $B$ relative to $A$? –  Qmechanic Jul 7 '13 at 9:21

2 Answers 2

Obviously, yess. Special Relativistic velocity addition formula:

$$v=\frac{v_1+v_2}{1+\frac{v_1v_2}{c_0^2}}$$

So, if $v_1$ is different, $v$ is also different. So, the spaceship will observe a different speed, of the flying garbage bin.

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The time inside spaceship would pass slower than Earth and distance measured by astronauts would be less than distance measured by Earth scientists. As $speed=distance/time$, they would observe different speed of something other than light. Speed of light would be observed same by all.

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