Question

1. A spaceship of mass m iş attached to a space station of mass M so that the Earth in a uniform circular orbit with orbital radius Ro. At one instant, the spac from the space station along the direction of motion. After the cjection, space station move in two different elliptical orbits around the Eart spaceship in its new elliptical orbit is 8R from the earth). In the figure, the dashed line is the original orbit before the ej shown line is the new orbit of the spaceship. The new orbit of so that they move around the eship is ejected the spaceship and the The apogce of the nal orbit before the ejection. The solid the space station is not shown. Spaceship 0 Space station Earth (a) Determine the orbital speed vo of the spaceship before the ejection in terms of Ro and the mass of the earth ME (b) Show that m vo-v where u and v are the velocities of the spaceship and space station right after the ejection respectively rght afer 4 (c) Let Rp be the perigee of the space station in its new elliptical orbit (i.e., the distance at which the station is closest to the Earth). If M 3m, determine Rp in terms of Ro

Answer 1

(a) for orbital speed . G (M + m) . Me/R_)^2 = (M + m) V_0^2/R_0 rightarrow V_0 = squareroot G Me/R_0 (b) Apply Conservation of momentum . (M + m) V_0 = m u + M . V rightarrow M (v_0 - v) = m (u . v_0) rightarrow if M = 3 m . 1/3 = V_0 v/u - v_0 rightarrow 3 v_0 - 3 v = u - v_0 4 v_0 = u + 3 v rightarrow (1) v = velocity of space station . for space ship . m . u . R_0 = m v_3 . 8 R_