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 $\frac { M } { 10 }$ 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): (1) $\frac { M } { 20 } \left( u ^ { 2 } + \frac { 113 } { 200 } \frac { G M _ { e } } { R } \right)$ (2) $5 M \left( u ^ { 2 } - \frac { 119 } { 200 } \frac { G M _ { e } } { R } \right)$ (3) $\frac { 3 M } { 8 } \left( u + \sqrt { \frac { 5 G M _ { e } } { 6 R } } \right) ^ { 2 }$ (4) $\frac { M } { 20 } \left( u - \sqrt { \frac { 2 G M _ { e } } { 3 R } } \right) ^ { 2 }$
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 $\frac { M } { 10 }$ 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):\\
(1) $\frac { M } { 20 } \left( u ^ { 2 } + \frac { 113 } { 200 } \frac { G M _ { e } } { R } \right)$\\
(2) $5 M \left( u ^ { 2 } - \frac { 119 } { 200 } \frac { G M _ { e } } { R } \right)$\\
(3) $\frac { 3 M } { 8 } \left( u + \sqrt { \frac { 5 G M _ { e } } { 6 R } } \right) ^ { 2 }$\\
(4) $\frac { M } { 20 } \left( u - \sqrt { \frac { 2 G M _ { e } } { 3 R } } \right) ^ { 2 }$