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## Homework Statement

It is required to put a satellite into an orbit with apogee of 5R/2, where R is the radius of the planet. The satellite is to be launched from the surface with a speed v

_{o}at 30

^{0}to the local vertical. If M is the mass of the planet show that v

_{0}=5GM/R (use conservation of energy and angular momentum). Assume that the planet is not rotating and that effects due to the planetary atmosphere can be ignored.

## Homework Equations

E= 1/2mv

^{2}+J

^{2}/(2mr

^{2})- GMm/r

J=mvr

l/r= 1+ecos(x)

## The Attempt at a Solution

i have attempted the question more than once, the following working is the attempt which i thought was closest to the answer.

rearranging the first equation gives

2E/m+2GM/r -J

^{2}/(m

^{2}r

^{2})=0

at r

_{max}, r=5R/2, dr/dt=0, x=pi

therefore

r

_{max}= -GMm/2E [ 1+ sqrt(1+2EL

^{2}/G

^{2}m

^{3}M

^{2})]

i dont know where to go from here.

2nd attempt:

v=v

_{0}sin30 (using launch angle)

therefore E= mv

_{0}

^{2}/4- GMm/r

E= -2.integral[Fdr] + 2GMm/r

therefore

2GM/r= v

_{0}

^{2}/4- GM/r

v

_{0}

^{2}= 12GM/r and r=5R/2

therefore

v

_{0}

^{2}= 24GM/5R