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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
  1. 1m
  2. 0.8m
  3. 0.5m
  4. 1.25m

A
1.25m
B
0.5m
C
1m
D
0.8m
Solution
Verified by Toppr

Given 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=mgh

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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