Question

Assume that a tunnel is dug across the earth $(radius=R)$ passing through its centre. Find the time a particle takes to reach centre of earth if it is projected into the tunnel from surface of earth with speed needed for it to escape the gravitational field of earth.

Answer 1

Lets consider a distance of $dr$ at a distance of $r$ from center of earth.
now, $dt=\frac{dr}{-v}$ [since the velocity is towards the center of earth]
Energy is conserved , Initial $=$ final energy
So, $-\frac{GMm}{R}+\frac{2mGM}{2}=\frac{m{v}^2}{2}+\frac{-GMm}{2R^3}(3R^2-r^2)$
solving this, we get $v={\left(\frac{GM(3R^2-r^2)}{R^3}\right)}^{0.5}$
$dt=\frac{dr}{-{\left(\frac{GM(3R^2-r^2)}{R^3}\right)}^{0.5}}$
Integrating on both sides with limits from $0$ to $R$
$T={\sin }^{-1}\left(\frac{1}{3^{0.5}}\right){\left(\frac{R^3}{GM}\right)}^{0.5}={\sin }^{-1}\left(\frac{1}{3^{0.5}}\right){\left(\frac{R}{g}\right)}^{0.5}$
So, $x=3$