A spacecraft which is moving with a speed u relative to the earth in the x -direction enters the gravitational field of a much more massive planet which is moving with a speed 3 u in the negative x -direction. The spacecraft exits following the trajectory as shown below. The speed of the spacecraft with respect to the earth a long time after it has escaped the planet's gravity is given by

Step-by-step solution

from momentum conservation - m u + m 1 3 u = m 1 v 1 + m v 2 … … 1 from energy conservation 1 2 m u 2 + 1 2 m 1 9 u 2 = 1 2 m 1 v 1 2 + 1 2 m v 2 2 1 2 m u 2 + 1 2 m 1 3 u - v 1 3 u + v 1 = 1 2 m v 2 2 from equation (1) ⇒ m 1 3 u - v 1 = m v 2 + u 1 2 m u 2 + 1 2 m v 2 + u 3 u + v 1 = 1 2 m v 2 2 as m 1 > > > m , we can assume v 1 ≈ 3 u u 2 + v 2 + u 6 u = v 2 2 ⇒ v 2 = 7 u