Question

A planet has mass $\dfrac{1}{10}th$ of that of Earth, while radius is $\dfrac{1}{3}rd$ of that of Earth. If a person can throw a stone on Earth surface to a height of $90m$, then he will be able to throw the stone on that planet to a height-
(A). $90m$
(B). $40m$
(C). $300m$
(D). $45m$

Answer 1

The force of gravity acting on the stone on Earth and the other planet will be different due to the difference in their masses and radii and hence acceleration due to gravity on their surfaces will also be different. Due to these factors the maximum height attained by the stone will also be different.

Formulae used:
$F=\dfrac{GMm}{{{R}^{2}}}$
$F=ma$
$P=mgh$

Complete step-by-step solution:
Every object exerts a force on another object; this force is known as the gravitational force. The gravitational force due to the Earth’s surface is given by-
$F=\dfrac{GMm}{{{R}^{2}}}$ (1)
Here,
$F$ is the gravitational force
$M$ is the mass of the Earth
$m$ is the mass of the body on or near the Earth’s surface
$R$ is distance between Earth and body
For objects on the Earth’s surface, $R$ is the radius of the Earth.

We know that, $F=ma$, substituting in eq (1), we get,
$\begin{aligned} & ma=\dfrac{GMm}{{{R}^{2}}} \\\ & \Rightarrow a=\dfrac{GM}{{{R}^{2}}} \\\ & \therefore g=\dfrac{GM}{{{R}^{2}}} \\\ \end{aligned}$
Here, $g$ is acceleration due to gravity

Given, a planet has mass $\dfrac{M}{10}$ and radius $\dfrac{R}{3}$, then acceleration due to gravity on its surface will be-
$\begin{aligned} & g'=\dfrac{G\dfrac{M}{10}}{\dfrac{R}{3}} \\\ & \Rightarrow g'=\dfrac{3}{10}\times \dfrac{GM}{R} \\\ & \therefore g'=\dfrac{3}{10}g \\\ \end{aligned}$
The other planet’s acceleration due to gravity is $\dfrac{3}{10}g$

When a stone is thrown on Earth’s surface it reaches a height of $90m$, its potential energy will be-
$\begin{aligned} & P=mgh \\\ & \Rightarrow P=m\times 10\times 90 \\\ & \therefore P=900m \\\ \end{aligned}$

Therefore, the maximum potential energy that can be stored in the stone is $900m\,J$. Using the potential energy, the maximum eight it can reach on the other planet is-
$\begin{aligned} & mg'h'=900m \\\ & \Rightarrow g'h'=900 \\\ & \Rightarrow \dfrac{3}{10}gh'=900 \\\ & \Rightarrow \dfrac{3}{10}\times 10h'=900 \\\ & \therefore h'=300m \\\ \end{aligned}$
Therefore, the maximum height which the stone can reach on the other planet is $300m$.

Hence, the correct option is (C).

Note:
The nature of gravitational force is always attractive. The energy stored in a body by virtue of its position is called its potential energy. Gravitational force exists between all bodies but its magnitude is very small, for larger bodies like Earth, due to their large masses, their force is prominent