JEE Main · 2019 · 08 Apr 2019 · Shift 2 · Gravitation

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A rocket has to be launched from earth in such a way that it never returns. If E is the minimum energy delivered by the rocket launcher, what should be the minimum energy that the launcher should have if the same rocket is to be launched from the surface of the moon? Assume that the density of the earth and the moon are equal and that the earth's volume is 64 times the volume of the moon.

(A) $\frac{E}{32}$
(B) $\frac{E}{16}$ ✓ Correct
(C) $\frac{E}{64}$
(D) $\frac{E}{4}$

Answer: (B)

Solution

The minimum launch energy equals the gravitational binding energy at the surface,

E = \frac{GM m}{R}

. Equal densities mean mass scales with volume, and the earth's volume is 64 times the moon's, so

R_{e a r t h} = 6 4^{1/3} R_{m oo n} = 4 R_{m oo n}

and

M_{e a r t h} = 64 M_{m oo n}

. Since the launch energy scales as

\frac{M}{R}

\frac{E _{m oo n}}{E _{e a r t h}} = \frac{M _{m oo n}}{M _{e a r t h}} \cdot \frac{R _{e a r t h}}{R _{m oo n}} = \frac{1}{64} \times 4 = \frac{1}{16}

. So from the moon the same rocket needs only

\frac{E}{16}

JEE Main · 2019 · 08 Apr 2019 · Shift 2 · Single correct option

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JEE Main · 2019 · 08 Apr 2019 · Shift 2 · Gravitation

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