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.
E= 1/2mv2 +J2/(2mr2)- GMm/r
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
at rmax, r=5R/2, dr/dt=0, x=pi
rmax= -GMm/2E [ 1+ sqrt(1+2EL2/G2m3M2)]
i dont know where to go from here.
v=v0sin30 (using launch angle)
therefore E= mv02/4- GMm/r
E= -2.integral[Fdr] + 2GMm/r
2GM/r= v02/4- GM/r
v02= 12GM/r and r=5R/2