# ESCAPE VELOCITY AND ORBITAL VELOCITY OF SATELLITE

Escape velocity: It is the minimum velocity with which a body must be protected from the surface of the earth so that it escapes from the gravitational field of the earth. We can also say that a body, projected with escape velocity, will be able to go to a point which distance from the earth.

Let the initial velocity of the particle be u. The kinetic energy of the particle is K = ½ mu2 and the gravitational potential energy of the earth particle system is U = – GMm/R where M is the mass of the earth, m is the mass of the particle and R is the radius of the earth. When it reaches a height h above the earth’s surface, its speed becomes v. The kinetic energy there is ½ mv2 and the gravitational potential energy is – GMm /R + h.

By conservation of energy $$\frac{1}{2}m{{u}^{2}}-\frac{GMm}{R}=\frac{1}{2}m{{v}^{2}}-\frac{GMm}{R+h}$$.

Or, $$\frac{1}{2}m{{v}^{2}}=\left\{ \frac{1}{2}m{{u}^{2}}-\frac{GMm}{R}~ \right\}+\frac{GMm}{R+h}$$ … (1)

The particle will reach the maximum height when v becomes zero.

If $$\frac{1}{2}m{{u}^{2}}-\frac{GMm}{R}~\ge 0$$, the right hand side of (1) is greater than zero for all values of h. Thus, ½ mv2 never becomes zero. The particle’s velocity never reaches zero and so the particle will continue to go farther and farther away from the earth. Thus, the particle will never return to the earth if

$$\frac{1}{2}m{{u}^{2}}-\frac{GMm}{R}~\ge 0$$.

Or, $$u~\ge \sqrt{\frac{2GM}{R}}$$.

This critical initial velocity is called the escape velocity. Putting the values of G, M & R the escape velocity from the earth comes out to be 11.6 km/s.

Orbital velocity: The velocity of a satellite in its orbit is called orbital velocity. Let v be orbital velocity of satellite, then

$$\frac{mv_{0}^{2}}{r}=\frac{GmM}{{{r}^{2}}}$$.

$${{v}_{0}}=\sqrt{\frac{GM}{r}}$$ or $${{v}_{0}}=\sqrt{\frac{GM}{R+h}}$$.

Hence orbital velocity decided by the radius of its orbit or its height above the earth surface

$${{v}_{0}}=\sqrt{\frac{GM}{r}}=\sqrt{gR}$$.

Time period of satellites: The time taken to complete revolution is called the time period. It is given by

$$T=\frac{2\pi r}{{{v}_{0}}}=2\pi r\sqrt{\frac{r}{GM}}$$.

$$T=\frac{2\pi {{r}^{3/2}}}{\sqrt{GM}}$$.

$${{T}^{2}}=\frac{4{{\pi }^{2}}}{GM}{{r}^{3}}$$.