Question

Ex. Find minimum speed with which particle should be projected so that it escapes to infinity

Answer 1

The minimum speed with which the particle should be projected so that it escapes to infinity from a distance $r$ from the center of a mass $M$ is given by: $v = \sqrt{\frac{2GM}{r}}$

If projected from the surface of a planet of mass $M$ and radius $R$ (i.e., $r=R$), the escape velocity is: $v_e = \sqrt{\frac{2GM}{R}} = \sqrt{2gR}$ (where $g$ is the acceleration due to gravity on the surface).

Answer 2

To escape to infinity, the particle must have just enough kinetic energy to overcome the negative potential energy due to the gravitational field. By applying the principle of conservation of mechanical energy, the total energy (kinetic + potential) of the particle at the starting point must be equal to its total energy at infinity. For minimum speed to escape, the particle's kinetic energy at infinity is zero. Equating the initial total energy $\frac{1}{2}mv^2 - \frac{GMm}{r}$ to the final total energy at infinity (which is zero, $0+0=0$), we solve for the initial speed $v$, which gives the escape velocity $\sqrt{\frac{2GM}{r}}$.