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Question

The maximum vertical distance through which a full dressed astronaut can jump on the Earth is 0.5 m. Estimate the maximum vertical distance through which astronaut can jump on the Moon, which has a mean density (23)rd that of the earth and radius one quarter that of the earth.

A
1.5 m
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B
3 m
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C
6 m
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D
5.5 m
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Solution

The correct option is B 3 m
The acceleration due to gravity at the surface of a planet is given by
g=GMR2g=GR2×43πR3ρ [M=43πR3ρ]

g=43πGρR

Hence,gρRLet, gm be the accelaration due to gravity at the surface of the moon and ge the accelaration due to gravity at the surface of the earth.

gmge=ρmρeRmRe

According to the given problem,

ρmρe=23 and RmRe=14

gmge=(23)(14)

gm=ge6

Since the astronaut has energy to jump only upto he=0.5 m on earth which will be equal to the work done by him to jump 0.5 m.

Since, he will do the same work on the both Earth and planet. So,

mgehe=mgphp

Substituting the values, in the above equation we get,

ge×0.5=ge6×hp

hp=3 m

Hence, option (b) is correct answer.

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