Question

The masses and radii of the earth and moon are M1, R1 and M2, R2 respectively. Their centres are at distance d apart. The minimum speed with which a particle of mass m should be projected from a point midway the two centres so as to escape to infinity is :-

Answer 1

Answer on Question #41093, Physics, Mechanics | Kinematics | Dynamics

Question:

The masses and radii of the earth and moon are M1, R1 and M2, R2 respectively. Their centres are at distance $d$ apart. The minimum speed with which a particle of mass $m$ should be projected from a point midway the two centres so as to escape to infinity is :-

Answer:

The law of conservation of energy:

$$T + U = \text{const}$$

where $T = \frac{mv^2}{2}$ is kinetic energy, $m$ - mass, $v$ - speed, $U = -\frac{GmM}{r}$ is potential energy, $r$ is distance to the center of second body.

Therefore:

$$\begin{array}{l} - \frac{G M_1 m}{\frac{d}{2}} - \frac{G M_2 m}{\frac{d}{2}} + \frac{m v^2}{2} = 0 + 0 + 0 \\ v^2 = \frac{4G}{d} (M_1 + M_2) \\ v = \sqrt{\frac{4G}{d} (M_1 + M_2)} \end{array}$$

Answer: $v = \sqrt{\frac{4G}{d} (M_1 + M_2)}$