A projectile is launched from the surface of a planet (mass = M,radius = R).What minimum launch speed is required if the projectile is to rise to a height of 2R above the surface of the planet? Disregard any dissipative effects of the atmosphere.
A) $\left[ \frac { 4 G M } { 3 R } \right] ^ { 1 / 2 }$
B) $\left[ \frac { 8 G M } { 5 R } \right] ^ { 1 / 2 }$
C) $\left[ \frac { 3 G M } { 2 R } \right] ^ { 1 / 2 }$
D) $\left[ \frac { 5 G M } { 3 R } \right] ^ { 1 / 2 }$
E) $\left[ \frac { G M } { 3 R } \right] ^ { 1 / 2 }$

Question

Quizplus · Accepted Answer

The minimum launch speed can be found using energy conservation: initial kinetic energy plus initial gravitational potential energy equals final gravitational potential energy. Solving for the initial speed gives $\left[ \frac { 4 G M } { 3 R } \right] ^ { 1 / 2 }$.

A Projectile Is Launched from the Surface of a Planet [4GM3R]1/2\left[ \frac { 4 G M } { 3 R } \right] ^ { 1 / 2 }[3R4GM​]1/2

A Projectile Is Launched from the Surface of a Planet $\left[ \frac { 4 G M } { 3 R } \right] ^ { 1 / 2 }$