Hello guys..!! Hope you all must be waiting eagerly for our next article. Here goes an explanation and derivation of escape velocity and its applications.

**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 us imagine what happens to a body of mass m if it is thrown from the earth with a velocity (escape velocity). As the body moves away from the earth, it’s slow sown hence, its kinetic energy is converted into gravitational potential energy of the mass-earth system.

KE lost by mass m=gain in gravitational potential energy of mass-earth system

\(\frac{1}{2}m{{v}_{e}}^{2}\,=\,\,{{U}_{f}}-{{U}_{i}}\).

\(\frac{1}{2}m{{v}_{e}}^{2}\,=\,\,0\,-\,\left( -\frac{GmM}{R} \right)\).

\({{v}_{e}}\,=\,\sqrt{\frac{2GM}{R}}\).

\({{v}_{e}}\,=\,\sqrt{2gR}\).

Substituting the values of g = 9.81m/s2 and R = 6400km we get v_{e} = 11.2 km/s

**Application of concepts of escape velocity:**

The following are the applications of the concept of escape velocity. The maximum velocity attained by a particle, orbital velocity and time period of satellites can be found if we know the escape velocity.

**(a) Maximum velocity attained by a particle: **Suppose a particle of mass m is projected vertically upwards with speed v and we want find the maximum height h attained by the particle. Then, we can use conservation of mechanical energy.

Decreases in kinetic energy=increases in gravitational potential energy

\(\frac{1}{2}m{{v}^{2}}\,=\,\frac{mgh}{1+\frac{h}{R}}\,\).

\(h=\frac{{{v}^{2}}}{2g-\frac{{{v}^{2}}}{R}}\).

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

\(\frac{m{{v}_{0}}^{2}}{r}\,=\,\frac{GmM}{{{r}^{2}}}\).

\({{v}_{0}}\,=\,\sqrt{\frac{GM}{r}}\).

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

\({{v}_{0}}\,=\,\sqrt{\frac{GM}{r}}\,=\,\sqrt{gR}\).

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

\(T\,=\,\frac{2\pi r}{{{v}_{0}}}\,\,=\,2\pi r\sqrt{\frac{r}{GM}}\).

\(T\,=\,\frac{2\pi {{r}^{3/2}}}{\sqrt{GM}}\).

\({{T}^{2}}\,=\,\frac{4{{\pi }^{2}}}{GM}{{r}^{3}}\).