Question

An object of mass $m$ is at rest on earth's surface. Escape speed of this object is $v_e$. Same object is orbiting earth with $h=R$, then its escape speed will be $v_e^{{}^{\prime }}$. Find the correct alternative

• A. $v_e^{{}^{\prime }}=\frac{v_e}{4}$
  
• B. $v_e^{{}^{\prime }}=2v_e$
  
• C. $v_e^{\prime }=\frac{v_e}{\sqrt{2}}$
  
• D. $v_e^{\prime }=\frac{\sqrt{2}{v}_e}{3}$
  

Answer 1

The correct option is C $v_e^{\prime }=\frac{v_e}{\sqrt{2}}$


The escape velocity for an object at a distance $r$ from the center of the Earth
$$v_e=\sqrt{\frac{2GM}{r}}$$On the surface of earth, $r=R$

$v_e=\sqrt{\frac{2GM}{R}}....\left(1\right)$

At the altitude $R$ from earth surface, $r=R+R=2R$

$v_e^{\prime }=\sqrt{\frac{2GM}{2R}}....\left(2\right)$

From equation $\left(1\right)$ and $\left(2\right)$,

$v_e^{{}^{\prime }}=\frac{v_e}{\sqrt{2}}$

Hence, option (c) is the correct answer.