SATELLITES

The term "satellite" refers to a body that is orbiting a larger body. A satellite is a body that revolves around a planet.

Natural satellites are objects that are naturally orbiting a planet.
For instance, the moon is the earth's natural satellite. The sun's satellite is the earth. To move in orbit around the target planets, man has also artificially placed certain satellites. These satellites are referred to as artificial satellites.

Escape velocity and Orbital velocity

Escape velocity is the smallest speed necessary to launch a body upward from the surface of the earth to escape gravity's influence.

Orbital velocity is the satellite's speed as it moves through its elliptical orbit around the planet.

Expression for Escape velocity $\left({\mathit{V}}_{\mathit{c}}\right)$

Consider a mass $\mathrm{m}$ body that is to be ejected at an escape velocity ${\mathrm{V}}_{\mathrm{e}}$ from the earth's surface. If the body is positioned on the planet's surface, then

Weight of body $=\mathrm{mg}$                                                                       $\left(1\right)$

Let $\mathrm{M}$ be the mass of the earth and $\mathrm{R}$ its radius. According to Newton’s law of gravitation

The gravitational force, $\mathrm{F}=\frac{\mathrm{GMm}}{{\mathrm{R}}^2}$                                                   $\left(2\right)$

where $\mathrm{G}$ is the gravitational constant.

The gravitational pull a body experiences when in the earth's atmosphere is proportional to its weight. Equating the equation $\left(1\right)$ and $\left(2\right)$ we get

Since, ${\mathrm{V}}_{\mathrm{e}}$ is the escape velocity given to the body

The given kinetic energy $=\frac{1}{2}{\mathrm{mV}}_{\mathrm{e}}^2$                                                      $\left(3\right)$ 

This kinetic energy is transformed into work to free the body from gravitational attraction.

If the body moves a distance $\mathrm{dR}$ in vertical direction against the force $\mathrm{F}$, then,

Work=Force x Distance

The effort put out by the body to travel from distance $\mathrm{R}$ to infinity in order to escape the gravitational pull of the universe is given by

Here, $\mathrm{GM}$$={\mathrm{gR}}^2$

Then, ${\mathrm{V}}_{\mathrm{e}}=\sqrt{\frac{2{\mathrm{gR}}^2}{\mathrm{R}}}$

Therefore, the Escape velocity ${\mathrm{V}}_{\mathrm{e}}=\sqrt{2\mathrm{gR}}$.

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