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 = %u2212 GMm/r2 where M is the mass of the Earth; the potential energy is %u2212GMm/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 please)

Explanation / Answer

conservation of angular momentum = > at largest diatane v*R = 1.3v1*r

conservation energy = > m ((1.3v1)^2 - v^2) /2 = GMm( (1/r) - (1/R) )

v = (1.3v1*r) /R = >

2GM(R-r ) = Rr(1.69v1^2 - 1.69 v1^2 *r^2/R^2)

2GM(R-r) R = r(R^2 - r^2) *1.69*v1^2

2GMR = r(R+r) 1.69*v1^2

and v1^2 = 2GM/r = >

R= (R+r )1.69

R= (1.69) *r/0.69 = 169r/69

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