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The radius in kilometer to which the present radius of earth $(R = 6400 km)$ to be compressed so that the escape velocity is increased 10 times is

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Solution

$Escape velocity v_{e} = \sqrt{\frac{2 G M}{R}}$
$\begin{array}{ll} \Rightarrow v_{e} \propto \frac{1}{\sqrt{R}} \\ \Rightarrow \frac{v_{1}}{v_{2}} = \sqrt{\frac{R_{2}}{R_{1}}} \\ \Rightarrow \frac{v}{10 v} = \sqrt{\frac{R_{2}}{R}} \\ \Rightarrow \frac{1}{100} = \frac{R_{2}}{R} \\ \Rightarrow R_{2} = \frac{R}{100} = \frac{6400}{100} = 64 k m \end{array}$

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Motion of planets and satellites

Remember concepts with our Masterclasses.

The radius in kilometer to which the present radius of earth R=6400 km to be compressed so that the escape velocity is increased 10 times is

Escape velocity ve=2GMR⇒ve ∝1R ⇒v1v2=R2R1 ⇒v10v=R2R⇒1100=R2R⇒R2=R100=6400100=64 km

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The radius in kilometer to which the present radius of earth $(R = 6400 km)$ to be compressed so that the escape velocity is increased 10 times is