Question

The escape velocity of a body on the surface of the earth is $11.2\text{km/s}$. If the earth's mass increases to twice its present value and radius of the earth becomes half, the escape velocity from the new surface becomes

• A. $22.4\text{km/s}$
• B. $44.8\text{km/s}$
• C. $5.6\text{km/s}$
• D. $11.2\text{km/s}$

Answer 1

The correct option is A $22.4\text{km/s}$

As we know that the escape velocity near the surface of earth is given by

$V_e=\sqrt{\frac{2GM}{R}}$

for new escape velocity,

$V_e^{\prime }=\sqrt{\frac{2G{M}^{\prime }}{R^{\prime }}}$

from question, $M^{\prime }=2M;R^{\prime }=R/2$

$V_e^{\prime }=\sqrt{\frac{2G\left(2M\right)}{\left(R/2\right)}}=2\sqrt{\frac{2GM}{R}}$

$\Rightarrow V_e^{\prime }=2V_e=22.4\text{km/s}$

Hence, option $\left(a\right)$ is correct.

$\begin{array}{c}\text{Why this question ?}\\ \text{To make students understand the}\\ \text{dependence of escape velocity on mass}\\ \text{and radius of a planet.}\end{array}$