A satellite starts in a circular orbit of radius r ( = distance from the center

Question

A satellite starts in a circular orbit of radius r ( = distance from the center of the Earth). Its speed is v1. The gravitational force on the satellite is central, Fr = ? GMm/r2 where M is the mass of the Earth; the potential energy is ?GMm/r. Suddenly, a burst of the rocket engine increases the speed of the satellite from v1 to v1+0.3 v1. Thereafter the satellite moves on an elliptical orbit, as shown.

Calculate the apogee ra (= greatest distance from the Earth). Express the answer in terms of the initial radius r. (Show your work!)

Explanation / Answer

the total energy in an elliptical orbit is given by (-GMm/2a) , where 2a is the distance between perigee and apogee => (1/2)mv^2 + (-GMm/r) = (-GMm/2a) => 1/2a = v^2/2GM - 1/r = (1.3v1)^2/2GM - 1/r => 1/2a = 1.69v1^2/2GM - 1/r => 2a = 2GMr/(1.69v1^2*r - 2GM) now, ra = 2a - r = 2GMr/(1.69v1^2*r - 2GM) - r = (4GMr - 1.69v1^2*r^2) / (1.69v1^2*r - 2GM)

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