To determine the speed at which the rocket must be fired relative to the spaceship, we can use the relativistic addition of velocities formula. This formula takes into account the effects of special relativity when adding velocities.
Let's denote:
v_r = velocity of the rocket relative to the spaceship
v_s = velocity of the spaceship relative to Earth
v_e = velocity of the rocket relative to Earth (0.854c in this case)
The relativistic addition of velocities formula is:
v_e = (v_r + v_s) / (1 + (v_r * v_s) / c^2)
Given that v_e = 0.854c and assuming that the spaceship's velocity v_s is negligible compared to the speed of light (c), we can simplify the equation to:
0.854c = v_r / (1 + (v_r * 0) / c^2)
0.854c = v_r / (1 + 0)
0.854c = v_r
Therefore, the rocket must be fired with a velocity of 0.854 times the speed of light (c) relative to the spaceship in order for it to travel at 0.854c relative to Earth.