Question

The radius in kilometers to which the present radius of the Earth to be compressed so that the escape velocity is increased $10$ times is

Answer 1

Step 1: Given Data

Let $G$ be the universal gravitational constant.

Let $M$ be the mass of Earth.

Let the initial radius of Earth be $R=6400km$.

Let the initial escape velocity be $v$.

The final escape velocity $v\text{'}=10v$

Step 2: Formula Used

Escape velocity $v=\sqrt{\frac{2GM}{R}}$

Step 3: Calculate the required Radius

We know that the escape velocity is given as,

$$\begin{gathered} v=\sqrt{\frac{2GM}{R}} \\ \Rightarrow v\sqrt{R}=const \end{gathered}$$

Therefore, if $R\text{'}$ is the new radius the second case should also be,

$v\text{'}\sqrt{R\text{'}}=const$

$$\begin{gathered} \therefore v\sqrt{R}=v\text{'}\sqrt{R\text{'}} \\ \Rightarrow v\sqrt{R}=10v\sqrt{R\text{'}} \\ \Rightarrow R=100R\text{'} \\ \Rightarrow R\text{'}=\frac{6400}{100} \\ \Rightarrow R\text{'}=64km \end{gathered}$$

Hence, the required radius of the Earth will be $64km$.