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.

We start by identifying the given information and assigning the proper signs: - The spaceship moves in the positive direction (away from Earth) at \[ v_{\text{ship/earth}} = +0.70\,c. \] - The ions are ejected from the back of the spaceship. Since the back is opposite to the forward (positive) direction, the speed of the ions relative to the spaceship is \[ v_{\text{ion/ship}} = -0.80\,c. \] The relativistic velocity addition formula is \[ v_{\text{ion/earth}} = \frac{v_{\text{ion/ship}} + v_{\text{ship/earth}}}{1 + \frac{v_{\text{ion/ship}} \, v_{\text{ship/earth}}}{c^2}}. \] Substitute the values: \[ v_{\text{ion/earth}} = \frac{(-0.80\,c) + (0.70\,c)}{1 + \frac{(-0.80\,c)(0.70\,c)}{c^2}}. \] Calculate the numerator: \[ -0.80\,c + 0.70\,c = -0.10\,c. \] Calculate the denominator: \[ 1 + \frac{(-0.80)(0.70)c^2}{c^2} = 1 - 0.56 = 0.44. \] Thus, \[ v_{\text{ion/earth}} = \frac{-0.10\,c}{0.44} \approx -0.2273\,c. \] The negative sign indicates that, with respect to the Earth, the ions are moving in the negative direction (i.e., toward the Earth).