Question

How much energy will be needed for a body of mass 100 kg to escape the Earth $(g = 10\,m/{s^2})$ given that the radius of Earth $= 6.4 \times {10^6}m$
A) $6.3 \times {10^9}$ Joules
B) $8 \times {10^9}$ Joules
C) $4 \times {10^9}$ Joules
D) ${\text{zero}}$ Joules

Answer 1

Use the law of conservation of energy when the body is on the surface of Earth and when it has escaped the gravity of Earth. The body will escape the Earth when it is infinitely far away from the Earth at which point, it will have no kinetic or potential energy.

Formula used:
-Potential energy due to gravity - $U = \dfrac{{ - GMm}}{R}$ where $G$ is the gravitational constant of acceleration, $M$ is the mass of the earth, $R$ is the distance of the object from Earth.

Complete step by step solution:
To determine the energy needed by the object to escape the gravity of Earth, we can use the law of conservation of energy. Thus the sum of initial potential and kinetic energy is equal to the sum of final kinetic and potential energy. So, we can write
$\Rightarrow {K_i} + {U_i} = {K_f} + {U_f}$
At the surface of the earth, the object will have kinetic energy ${K_i}$ and potential energy corresponding to the radius or the earth ${U_i} = - \dfrac{{GMm}}{R}$
Once the object escapes the Earth, it will be at an infinite distance away from the Earth and won’t have any kinetic or potential energy so ${K_f} = {U_f} = 0$. Substituting these values in equation (1), we get
$\Rightarrow {K_i} - \dfrac{{GMm}}{R} = 0$
$\Rightarrow {K_i} = \dfrac{{GMm}}{R}$
Since we haven’t been given the mass of Earth we can instead use the relation of gravitational acceleration on the surface of the Earth as
$\Rightarrow g = \dfrac{{GM}}{{{R^2}}} \Rightarrow GM = g{R^2}$
Substituting the value of $GM$ in equation (2), we get
$\Rightarrow {K_i} = \dfrac{{(g{R^2})m}}{R} = gmR$
Substituting the value of $g$, $m = 100kg$ and $R = 6.4 \times {10^6}m$ , we get
$\Rightarrow {K_i} = 10 \times 100 \times 6.4 \times {10^6}$
$\Rightarrow = 6.4 \times {10^9}\,{\text{Joules}}$
Hence we require $= 6.4 \times {10^9}\,{\text{Joules}}$ of energy for the body to escape the Earth.

Note:
To solve such problems we only need to compare the kinetic and potential energies of the initial and final conditions of the system. While we have assumed here that the object will have to go an infinite distance to escape the Earth, in reality, gravity due to other objects in space will overcome the gravity due to Earth, and then the object can be said to have escaped the Earth.