A rocket is launched vertically from the surface of the earth of radius R with an initial speed v. If atmospheric resistance is neglected, the maximum height attained by the rocket is given by
h=R(2gRv2−1)
On the surface of the earth, the total energy is
KEi+PEi=12mv2−GmMR
where m is the mass of the rocket and M that of the earth. At the highest point, v = 0 and the energy is entirely potential.
PEf=−GmM(R+h) where h is the maximum height attained.
From the law of conservation of energy, we have
12mv2−GmMR=−GmM(R+h)
⇒R+hh=2gRv2(∵g=GMR2)
⇒h=R(2gRv2−1)
Hence, the correct choice is (a).