Question

A particle of mass $m$ is kept at rest at a height $3R$ from the surface of earth , where ${}^{\prime }{R}^{\prime }$ 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

• A. ${\left(\frac{GM}{2R}\right)}^{1/2}$
• B. ${\left(\frac{gM}{4}\right)}^{1/2}$
• C. ${\left(\frac{2g}{R}\right)}^{1/2}$
• D. ${\left(\frac{GM}{R}\right)}^{1/2}$

Answer 1

The correct option is A ${\left(\frac{GM}{2R}\right)}^{1/2}$

As we know, the minimum speed with which a body is projected so that it does not return back is called escape speed.

$v_e=\sqrt{\frac{2GM}{r}}$ where $r=R+h$

$=\sqrt{\frac{2GM}{R+h}}$ where $h=3R$

$=\sqrt{\frac{2GM}{4R}}$

$=\sqrt{\frac{GM}{2R}}$

$={\left(\frac{GM}{2R}\right)}^{\frac{1}{2}}$