Question

The escape velocity for a planet is $v_e$. A tunnel is dug along a diameter of the planet and a small body is dropped into it at the surface. When the body reaches the center of the planet, its speed will be

• A. $v_e$
• B. $\frac{v_e}{\sqrt{2}}$
• C. $\frac{v_e}{2}$
• D. Zero

Answer 1

The correct option is B $\frac{v_e}{\sqrt{2}}$

Let the mass of the body be $m$
$v_e=\sqrt{\frac{2GM}{R}}$........(1)
Since initial velocity is zero, therefore initial energy
$E_i=\frac{-GMm}{R}$.............(2)
Final Energy, $E_f=\frac{-GMm(3R^2-r^2)}{2R^3}+\frac{m{v}^2}{2}$...........(3)
Since total mechanical energy is conserved, therefore (1) and (2) are equal. Also, $r=0$.
Equating (2) and (3) we get
$v=\sqrt{\frac{GM}{R}}$......(4)
Comparing equation (1) and (4), we get
$v=\frac{v_e}{\sqrt{2}}$