Question

How does the escape velocity of a body vary with the mass of the earth?

Answer 1

Dear Student ,
We know from the theory of gravitation that escape velocity of a body is , $v_e=\sqrt{2g{R}_{earth}}$
Now again we know that the gravitational acceleration of a body on earth is , $g=\frac{G{M}_{earth}}{{R_{earth}}^2}$
So putting the value of 'g' in the equation of escape velocity v_e we get ,
$$\begin{gathered} v_e=\sqrt{2\times \frac{G{M}_{earth}}{{R_{earth}}^2}\times R_{earth}}=\sqrt{\frac{2G{M}_{earth}}{R_{earth}}} \\ So,escapevelocity{v}_e\propto \sqrt{M_{earth}} \\ Thereforeescapevelocityofabodyvarywiththesquarerootofthemassoftheearth. \end{gathered}$$
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