Observer on earth sees a ship travelling at \( 0.5 \mathrm{c}[\mathrm{fwd}] \) wrt the ground. A plasma cannon fires a pulse out the front of the ship so that the ground observer sees the pulse travel at 0.9 c [fwd] wrt the ground. How fast do the observers in the ship see the pulse travel wrt the ship?

To find the speed of the pulse as observed from the ship, we can use the relativistic velocity addition formula. This formula states that if an object (like the pulse) moves at velocity \( u' \) in a frame moving at velocity \( v \), the velocity \( u \) in the stationary frame is given by: \[ u = \frac{u' + v}{1 + \frac{u'v}{c^2}} \] In this scenario: - The ship's speed \( v = 0.5c \) - The pulse speed relative to the ground \( u = 0.9c \) We need to find \( u' \) (the speed of the pulse relative to the ship). Using the rearranged formula: \[ u' = \frac{u - v}{1 - \frac{uv}{c^2}} \] Substituting the values: \[ u' = \frac{0.9c - 0.5c}{1 - \frac{(0.9c)(0.5c)}{c^2}} = \frac{0.4c}{1 - 0.45} = \frac{0.4c}{0.55} \approx 0.727c \] Thus, the observers in the ship see the pulse travel at approximately \( 0.727c \) with respect to the ship.