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 ve = 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
\(T\,=\,\frac{2\pi r}{{{v}_{0}}}\,\,=\,2\pi r\sqrt{\frac{r}{GM}}\).
\(T\,=\,\frac{2\pi {{r}^{3/2}}}{\sqrt{GM}}\).
\({{T}^{2}}\,=\,\frac{4{{\pi }^{2}}}{GM}{{r}^{3}}\).