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clarabelle
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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 vo at 300 to the local vertical. If M is the mass of the planet show that v0=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= 1/2mv2 +J2/(2mr2)- 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 -J2/(m2r2)=0
at rmax, r=5R/2, dr/dt=0, x=pi
therefore
rmax= -GMm/2E [ 1+ sqrt(1+2EL2/G2m3M2)]
i don't know where to go from here.
2nd attempt:
v=v0sin30 (using launch angle)
therefore E= mv02/4- GMm/r
E= -2.integral[Fdr] + 2GMm/r
therefore
2GM/r= v02/4- GM/r
v02= 12GM/r and r=5R/2
therefore
v02= 24GM/5R