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a projectile is fired from the earth surface of radius R with velocity "kv"(v=escape velocity from the earth surface and k<1). Neglecting air resistance, the maximum height of rise from the centre of the earth is ------------
initial PE = -GMm/R
(KE)i = mV2/2 = mk2v2/2
v = (2GM/R)1/2 (escape speed = (2GM/R)1/2)
so , (KE)i = k2GMm/R
total energy initial = GMmK2/R - GMm/R ......................1
finally particle comes to rest after attaining a H height...
now PE = -GMm/(R+H)
(KE)f = 0
total energy = -GMm/(R+H) .................2
since mechanical energy is conserved so eq1 = eq2
-GMm/(R+H) = -GMM/R +K2GMm/R
H/R+H = K2
H = RK2/(1-K2)
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