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Gravitational Potential Energy

The velocity ...

Question

The velocity with which it must be projected is $\sqrt{\frac{2 n g R}{n + 1}}$ ,where $R$ is the radius of the earth and $m$ the mass of body.

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Solution

The change in potential energy must be equal to the initial kinetic energy since final kinetic energy is 0.
Hence,
$Δ E = \frac{n}{n + 1} m g R = \frac{1}{2} m v^{2}$
$v = \sqrt{\frac{2 n g R}{n + 1}}$
Answer is $\sqrt{\frac{2 n g R}{n + 1}}$

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