MHT CET · Physics · Gravitation

What is the minimum energy required to launch a satellite of mass ‘m’ from the surface of the earth of mass ‘M’ and radius ‘R’ at an attitude 2R?

A $\frac{G M m}{2 R}$
B $\frac{2 G M m}{3 R}$
C $\frac{G M m}{3 R}$
D $\frac{5 G M m}{6 R}$

Verified Solution

Answer & Solution

Correct Answer

(B) $\frac{2 G M m}{3 R}$

Step-by-step Solution

Detailed explanation

The minimum energy required to launch a satellite from surface of earth is equal to the amount of work done against gravity to move it from the surface of earth to a height h and is given by

U = W = G M m (\frac{1}{r_{1}} - \frac{1}{r_{2}})

= G M m (\frac{1}{R} - \frac{1}{R + h})

Here, h = 2R

\Rightarrow U = G M m (\frac{1}{R} - \frac{1}{3 R}) = \frac{2 G M m}{3 R}

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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