Question

The escape speed of a projectile on the earth’s surface is 11.2 km 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, $v_{esc}$ = $11.2 \; km/s$
Projection velocity of the projectile, $v_p$ = $3v_{esc}$
Mass of the projectile = $m$
Velocity of the projectile far away from the Earth = $v_f$
Total energy of the projectile on the Earth = $\frac{1}{2}mv^2_p - \frac{1}{2}mv^2_{esc}$
Gravitational potential energy of the projectile far away from the Earth is zero.
Total energy of the projectile far away from the Earth= $\frac{1}{2}mv^2_f$

From the law of conservation of energy, we have
$\frac{1}{2}mv^2_p - \frac{1}{2}mv^2_{esc}$ = $\frac{1}{2}mv_f^2$

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

=$\sqrt{(3v_{esc}) ^2 - (v_{esc})^2}$

=$\sqrt 8 v_{esc}$

=$\sqrt 8 \times 11.2$ = $31.68 \; km/s$