Question

The value of $ g $ on moon is $ 1/6 $ th of the value of $ g $ on earth. A man can jump $ 1.5m $ high on the earth. He can jump on the moon up to a height of:
(A) $ 9m $
(B) $ 7.5m $
(C) $ 6m $
(D) $ 4.5m $

Answer 1

Hint : To solve this question, we have to find the potential energy gained by the man from the two surfaces of the earth and the moon. Then using the energy conservation principle this energy is equal to the maximum potential energy of the man. Equating these energies on the two surfaces, we will get the final answer.

Formula Used: In this solution we will be using the following formula,
 $ U = mgh $ where $ U $ is the potential energy, $ m $ is the mass, $ g $ is the acceleration due to gravity and $ h $ is the height.

Complete step by step answer
We are able to jump because of the reaction force offered by the surface just below us. When we are standing still on the ground, then this reaction force equals our weight. But when we make a jump, we basically apply a force on the ground by the muscles of our leg, in addition to our weight, so by Newton's third law of motion, the ground applies an equal amount of the reaction force. Since our downward force on the surface is greater than our weight, so the reaction force applied upwards by the surface is also greater than our weight. So with this reaction force we are able to accelerate upwards. We basically gain kinetic energy to move upwards, and at the height when the gravitational potential energy equals this energy, we begin falling downwards. So the kinetic energy gained by us from the surface is equal to the maximum gravitational potential energy.
Let the mass of the man be $ m $ . According to the question, the man can jump $ 1.5m $ high on the earth. So the energy gained by the man from the earth’s surface is
 $ {U_1} = mg\left( {1.5} \right) $ ......................(1)
Now, let the maximum height reached by the man on the moon be $ h $ . According to the question the acceleration due to gravity on the moon is $ \dfrac{g}{6} $ . So the kinetic energy gained by the man from the moon’s surface is
 $ {U_2} = m\dfrac{g}{6}h $ ......................(2)
Equating (1) and (2) we have
 $ mg\left( {1.5} \right) = m\dfrac{g}{6}h $
Dividing both sides by $ mg $ we get
 $ 1.5 = \dfrac{h}{6} $
 $ \Rightarrow h = 9m $
Thus the man is able to jump on the moon up to a height of $ 9m $ .

Note
The energy which a person can gain from a surface depends on his strength. It is independent of the surface on which he is standing. That is why we were able to equate the energies on the two surfaces.