Question 346: Gravitation

Question

The masses and radii of the moon and the earth are $$\mathrm{M_1, R_1}$$ and $$\mathrm{M_2, R_2}$$ respectively. Their centres are at a distance $$\mathrm{d}$$ apart. What should be the minimum speed with which a body of mass '$$m$$' should be projected from a point midway between their centres, so as to escape to infinity?

Accepted Answer

$$\mathrm{O}$$ is the midpoint of the line joining the centres of $$\mathrm{A}$$ and $$\mathrm{B}$$. and a body (C) of mass '$$\mathrm{m}$$' is kept at $$\mathrm{O}$$
The P.E. of $$\mathrm{C}$$ is
$$\mathrm{U}=-\frac{\mathrm{GM}_1 \mathrm{~m}}{\mathrm{~d} / 2}-\frac{\mathrm{GM}_2 \mathrm{~m}}{\mathrm{~d} / 2}=-\frac{2 \mathrm{Gm}}{\mathrm{d}}\left(\mathrm{M}_1+\mathrm{M}_2ight)$$
Initially, $$\mathrm{C}$$ is at rest, its K.E. $$=0$$
$$	herefore$$ Total energy of $$\mathrm{C}=-\frac{2 \mathrm{Gm}}{\mathrm{d}}\left(\mathrm{M}_1+\mathrm{M}_2ight)$$
$$	herefore$$ Its binding energy $$=\frac{2 \mathrm{GM}}{\mathrm{d}}\left(\mathrm{M}_1+\mathrm{M}_2ight)\quad 	ext{..... (1)}$$
Let $$v_e$$ be the velocity that should be given to the body to escape to infinity.
For this its K.E. = Binding energy
$$\begin{aligned}
& 	herefore \frac{1}{2} \mathrm{mv}_{\mathrm{e}}^2=\frac{2 \mathrm{Gm}}{\mathrm{d}}\left(\mathrm{M}_1+\mathrm{M}_2ight) \\
& 	herefore \mathrm{v}_{\mathrm{e}}^2=\frac{4 \mathrm{G}\left(\mathrm{M}_1+\mathrm{M}_2ight)}{\mathrm{d}} \\
& 	herefore \mathrm{v}_{\mathrm{e}}=2 \sqrt{\frac{\mathrm{G}\left(\mathrm{M}_1+\mathrm{M}_2ight)}{\mathrm{d}}}
\end{aligned}$$

Answer

$$\frac{\mathrm{G}\left(\mathrm{M}_1+\mathrm{M}_2\right)}{\mathrm{d}}$$

Answer

$$\sqrt[2]{\frac{G\left(M_1+M_2\right)}{d}}$$

Answer

$$\sqrt{\frac{G d}{M_1+M_2}}$$

Answer

$$\sqrt{\frac{M_1+M_2}{G d}}$$

Gravitation

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