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Question

The ratio of radii of the earth to another planet is 23 and the ratio of their mean densities is 45. If an astronaut can jump to a maximum height of 1.5m on the earth, with the same effort, the maximum height he can jump on the planet is

A
1m
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B
0.8m
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C
0.5m
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D
1.25m
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Solution

The correct option is B 0.8mGiven parameters: RR′=23 ρρ′=45 Maximum height on the earth, he=1.5m Mass of the earth will be: Me=ρ×V Me=ρ×43πR3 Gravitational acceleration at earth's surface will be - g=GMeR2 g=G×ρ×43πR3R2 g=G×ρ×43πR Similarly, gravitational acceleration at the given planet's surface will be: g′=G×ρ′×43πR′ Let E be common energy given to astronaut of mass m on both planets. At the maximum height, this energy will be converted to potential energy. So, mgh=mg′h′ i.e.h′=hgg′ Putting values of g and g’ , we get h′=hG×ρ×43πRG×ρ′×43πR′ h′=h×ρρ′×RR′ h′=1.5×45×23 h=0.8 m Therefore, the astronaut can jump to max height of 0.8 m. The correct option is B.

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