MHT CET · Physics · Gravitation

Earth has mass ' \(M_1\) ' radius ' \(R_1\) ' and for moon mass ' \(\mathrm{M}_2\) ' and radius ' \(\mathrm{R}_2\) '. Distance between their centres is ' \(r\) '. A body of mass ' \(M\) ' is placed on the line joining them at a distance \(\frac{r}{3}\) from the centre of the earth. To project a mass ' M ' to escape to infinity, the minimum speed required is

A \(\left[\frac{2 G}{r}\left(M_2+\frac{M_1}{2}\right)\right]^{1 / 2}\)
B \(\left[\frac{4 G}{r}\left(M_1+\frac{M_2}{2}\right)\right]^{1 / 2}\)
C \(\left[\frac{3 G}{r}\left(M_1+M_2\right)\right]^{1 / 2}\)
D \(\left[\frac{6 \mathrm{G}}{\mathrm{r}}\left(\mathrm{M}_1+\frac{\mathrm{M}_2}{2}\right)\right]^{1 / 2}\)

Verified Solution

Answer & Solution

Correct Answer

(D) \(\left[\frac{6 \mathrm{G}}{\mathrm{r}}\left(\mathrm{M}_1+\frac{\mathrm{M}_2}{2}\right)\right]^{1 / 2}\)

Step-by-step Solution

Detailed explanation

The binding energy of the body is given by
\(\begin{aligned}
\text { B.E. } & =\frac{\mathrm{GM}_1 \mathrm{M}}{\frac{\mathrm{r}}{3}}+\frac{\mathrm{GM}_2 \mathrm{M}}{\frac{2 \mathrm{r}}{3}}=\frac{3 \mathrm{GM}_1 \mathrm{M}}{\mathrm{r}}+\frac{3 \mathrm{GM}_2 \mathrm{M}}{2 \mathrm{r}} \\
& =\frac{3 \mathrm{GM}}{\mathrm{r}}\left[\mathrm{M}_1+\frac{\mathrm{M}_2}{2}\right]
\end{aligned}\)
If \(v\) is the velocity given to the body, then
\(\begin{aligned}
& \frac{1}{2} M v^2=\frac{3 G M}{r}\left[M_1+\frac{M_2}{2}\right] \\
\therefore \quad & v=\left[\frac{6 G}{r}\left(M_1+\frac{M_2}{2}\right)\right]^{\frac{1}{2}}
\end{aligned}\)

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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