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)

Approved