Question

A particle of mass $m$ is kept at rest at a height $3R$ 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)

• A. $\sqrt{\left(\frac{GM}{R}\right)}$
• B. $\sqrt{\left(\frac{GM}{2R}\right)}$
  
• C. $\sqrt{\left(\frac{2GM}{R}\right)}$
• D. $\sqrt{\left(\frac{GM}{3R}\right)}$

Answer 1

The correct option is B $\sqrt{\left(\frac{GM}{2R}\right)}$


As we know, the minimum speed with which a body is projected so that it does not return back is called escape speed, which is given by$$v_e=\sqrt{\frac{2GM}{r}}$$Where $r$ is the distance from the centre of the earth,
given, $r=R+3R=4R$

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

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

Hence, option (b) is the correct answer.

Why this question: Escape velocity can be calculated for any orbit not just the surface of the planet. For a particle to escape its total mechanical energy should be equal to zero.