Question

A rocket is launched from a spaceship travelling with a speed of 0.8c towards the
earth. If the speed of the rocket is 0.6c, what would be its speed as observed by an
observer on earth if the rocket is moving
(i) towards the earth (ii) away from the earth?

Answer 1

The speeds of a spaceship and a rocket are comparable to the speed of light, so we must use the formula for adding the velocities in the relativistic case. For the case of movement along the same direction, this formula can be written as $V_x=c\frac{\beta_{sp} + \beta_{r}}{1+\beta_{sp} \beta_{r}}$ where $\beta_{sp}=\frac{V_{sp}}{c}$ $\beta_{r}=\frac{V_{r}}{c}$ spaceship and rocket speed related с. We have for case (i)

$V_i=c\frac{0.8+0.6}{1+0.8\cdot 0.6}=0.946c$ . For the case (ii) we have $\beta_{r}=-0.6$ and

$V_{ii}=c\frac{0.8-0.6}{1-0.8\cdot 0.6}=0.385c$

Answer:  If the rocket is moving

(i) towards the earth its speed relative to the Earth is 0.946c (ii) away from the earth its speed relative to the Earth is 0.385c