Question

The escape speed of a projectile on the earth's surface is $11.2km{s}^{-1}$. A body is projected out with thrice this speed. What is the speed of the body far away from the earth? Ignore the presence of the sun and other planets.

Answer 1

Escape velocity of a projectile from the Earth, $v_{esc}=11.2km/s$
Projection velocity of the projectile, $v_p=3v_{esc}$
Mass of the projectile $=m$
Let velocity of the projectile far away from the Earth$=v_f$
Total energy of the projectile on the Earth $=\frac{1}{2}m{v}_p^2-\frac{1}{2}m{v}_{esc}^2$

Gravitational potential energy of the projectile far away from the Earth is zero.
Total energy of the projectile far away from the Earth $=\frac{1}{2}m{v}_f^2$
From the law of conservation of energy, we have
$\frac{1}{2}m{v}_p^2-\frac{1}{2}m{v}_{esc}^2=\frac{1}{2}m{v}_f^2$

$v_f=\sqrt{v_p^2-v_{esc}^2}$

$=\sqrt{8}{v}_{esc}$
$=\sqrt{8}\times 11.2$
$=31.68km/s$