Question

A satellite of mass m is traveling at speed vo in a circular orbit of radius ro under the gravitational force of a fixed mass M at point O. At a certain point Q in the orbit (see the figure below) the direction of motion of the satellite is suddenly changed by an angle α without any change in the magnitude of the velocity. As a result the satellite goes into an elliptic orbit. Its distance of the closest approach to O (at point P) is ro/5. 1. What is the speed of the satellite at P, expressed as a multiple of v? 2. Find the angle a. 个015

Answer 1

Orbital velocity of satellites is v_0 = squareroot GM/r_0 let speed of satellite at p is v By angular momentum conservation at Q and p we get mv_0 cos (alpha)r_0 = mvr_0/5 equation 1 v = 5v_0 cos alpha equation 1 By mechanical conservation between Q + P we get -GMm/r_0 + 1/2 mv^2_0 = -5GMm/r_0 + 1/2 mv^2 mv^2/2 = 4GMm/r_0 + mv^2_0/2 v^2 = 8GM/r_0 + v^2_0 v^2 = 8v^2_0 + v^2_0 v = 3v_0 from equation 1 cos alpha = v/ 5v_0 = 3v_0/5v_0 alpha = cos^- 1 (3/5)