A satellite of mass M is launched vertically upwards with an initial speed u from the surface of the earth. After it reaches height R ( R = radius of the earth), it ejects a rocket of mass M 10 so that subsequently the satellite moves in a circular orbit. The kinetic energy of the rocket is ( G is the gravitational constant; M e is the mass of the earth):

Step-by-step solution

- GM e M R + 1 2 Mu 2 = - GM e M 2 R + 1 2 Mv 2 v = u 2 - G M e R V τ → Transverse velocity of rocket V R → Radial velocity of rocket V = G M e 2 R M 10 V T = 9 M 10 G M e 2 R M 10 V r = M u 2 - GM e R Kinetic energy = 1 2 M 10 V T 2 + V r 2 = M 20 81 GM e 2 R + 100 u 2 - 100 GM e R = M 20 100 u 2 - 119 G M e 2 R = 5 M u 2 - 119 GM e 200 R