Q157 / 180 · COMEDK 2013 · Physics
A particle of mass $m$ is kept at rest at a height $3 R$ from the surface of earth, where $R$ is radius of earth and $M$ is mass of earth. The minimum speed with which it should be projected, so that it does not return back, is ( $g$ is acceleration due to gravity on the surface of earth)
- Correct answerA$\left(\frac{G M}{2 R}\right)^{1 / 2}$
- B$\left(\frac{g R}{4}\right)^{1 / 2}$
- C$\left(\frac{2 g}{R}\right)^{1 / 2}$
- D$\left(\frac{G M}{R}\right)^{1 / 2}$
Solution
The escape velocity from earth's surface is given by
$$
\begin{aligned}
v_{e} &=\sqrt{\frac{2 G M}{R+h}} \\
\text { Given, } \quad h &=3 R \\
\Rightarrow \quad v_{e} &=\sqrt{\frac{2 G M}{4 R}}=\sqrt{\frac{G M}{2 R}}
\end{aligned}
$$
Hence, the correct answer is option: 'A'.