Two identical spherically symmetric planets, each of mass M, are somehow held at rest with respect to each other. Each planet has radius R, and the distance between the centers of the planets is 4R. If a rocket is launched from the surface of one planet w

Question

Two identical spherically symmetric planets, each of mass M, are somehow held at rest with respect to each other. Each planet has radius R, and the distance between the centers of the planets is 4R. If a rocket is launched from the surface of one planet with speed v, what is the minimum speed v so that the rocket can reach the other planet?

Smart Achievers · Accepted Answer

(4)

The gravitational force vanishes at the midway point between the planets, so the rocket only needs to have enough energy to get there. The initial and final gravitational potential energies are $U_{i} = - \frac{G M m}{R} - \frac{G M m}{3 R} = - \frac{4 G M m}{3 R}$ and $U_{f} = - \frac{2 G M m}{2 R} = - \frac{G M m}{R}$

Solution

1 $\sqrt{\frac{2 G M}{R}}$

2 $\sqrt{\frac{G M}{R}}$

3 $\sqrt{\frac{3 G M}{4 R}}$

4 $\sqrt{\frac{2 G M}{3 R}}$

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Solution

1 2GMR

2 GMR

3 3GM4R

4 2GM3R

Ans. (4) The gravitational force vanishes at the midway point between the planets, so the rocket only needs to have enough energy to get there. The initial and final gravitational potential energies are Ui=-GMmR-GMm3R=-4GMm3R and Uf=-2GMm2R=-GMmR

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1 $\sqrt{\frac{2 G M}{R}}$

2 $\sqrt{\frac{G M}{R}}$

3 $\sqrt{\frac{3 G M}{4 R}}$

4 $\sqrt{\frac{2 G M}{3 R}}$