Question

Show the nature of the following graph for a satellite orbiting the earth.
(a) `K E` vs orbitial radius `R` (b) `PE` vs orbital radius `r` (c ) `TE` vs orbital radius `R`.

Answer 1

Consider the diagram, where a satellite of mass `m`, moving around the earth in a circular orbit of radius R.

Orbital speed of the satellite orbitting the earth is given by `v_(0) = sqrt((GM)/(R))`
where, M and R are the mass and radius of the earth
(a) `:.` KE of a satellite of mass m, `E_(K) = (1)/(2) mv_(0)^(2) = (1)/(2) m xx (GM)/(R)`
`:. E_(K) prop (1)/(R)`
It means the KE decreases exponentially with radius.
The graph for KE versus orbital radius R is shown in figure.

(b) Potential energy of a satellite `E_(P) = - (GMm)/(R)`
`E_(P) prop - (1)/(R)`
The graph for PE versus orbital radius R is shwon in figure.

(c) Total energy of the satellite `E = E_(K) + E_(P) = (GMm)/(2R) - (GMm)/(R)`
`= - (GMm)/(2R)`
The graph for total energy versus orbital radius R is shown in the figure.

Note. We should keep in mind that Potential Energy (PE) and Kinetic Energy (KE) of the satellite - earth system is always negative.