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A man can jump 1.5m high on earth. He ca...

A man can jump `1.5m` high on earth. He can jump on a planet to a height of `3 xx x` metre. The density of planet is one quarter that of the earth and whose radius is one third of the earth. What is the value of `x`?

Text Solution

Verified by Experts

The correct Answer is:
`(6)`
Let `rho` and `rho'` be the density of earth and planet respectively `R` and `R'` be their radii, then `rho = 4 rho'` and `R = 3R'`. Let `h` and h' be the height of jump of a man on earth and planet respectively. `g` and g' be the acceleration due to gravity on earth and planet respectively. Here , `h = 1.5 m , h' = 3 xx xm`
`g = (GM)/(R^(2)) = (G)/(R^(2)) xx [(4)/(3) pi R^(3) rho] = (4)/(3) pi GR rho`
and `g' = (GM')/(R'^(2)) = (G)/(R^(2)) [(4)/(3) pi R'^(3) rho'] = (4)/(3) pi GR' rho'`
The gain in potential energy at the higher point will be the same in both the cases. Hence
`mg' h' = mgh` or `h' = gh//g'`
or `h = (((4)/(3) pi GR rho))/(((4)/(3) pi GR' rho')) xx h = (R)/(R') xx (rho)/(rho') xx h`
`= (3R')/(R') xx (4rho')/(rho') xx 1.5 = 18 = 3 xx x` (given)
or `x = 6`
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