7. Spaceships of the future may be powered by ion-propulsion engines in which ions are ejected from the back of the ship to drive it forward. In one such engine the ions are to be ejected with a speed of 0.80 c relative to the spaceship. The spaceship is traveling away from the earth at a speed of 0.70 c relative to the earth. What is the velocity of the ions relative to the earth? Assume that the direction in which the spaceship is traveling is the positive direction, and be sure to assign the correct plus or minus signs to the velocities.

To find the velocity of the ions relative to the Earth, we can use the relativistic velocity addition formula: \[ v_{final} = \frac{v_1 + v_2}{1 + \frac{v_1 v_2}{c^2}} \] Here, \(v_1\) is the speed of the spaceship relative to Earth (0.70c) and \(v_2\) is the speed of the ions relative to the spaceship (-0.80c, as they are ejected backward). Substituting the values into the formula: \[ v_{final} = \frac{0.70c + (-0.80c)}{1 + \frac{(0.70)(-0.80)c^2}{c^2}} \] \[ v_{final} = \frac{0.70c - 0.80c}{1 - 0.56} = \frac{-0.10c}{0.44} \approx -0.227c \] Thus, the velocity of the ions relative to Earth is approximately \(-0.227c\), indicating that the ions are moving backward relative to Earth at about 22.7% of the speed of light.