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Question
Show the nature of the following graph for a satellite orbiting the earth.
- KE vs orbital radius R
- PE vs orbital radius R
- TE vs orbital radius R.
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Solution
Consider the diagram, where a satellite of mass m, moves around the earth in a circular orbit of radius R.
The orbital speed of the satellite orbiting 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,
= `1/2 mv_0^2 `
= `1/2m xx (GM)/R`
∴ `E_k ∝ 1/R`
It means the KE decrease exponentially with radius. The graph for KE versus orbital radius R is shown in figure.
b. Potential energy of a satellite `E-p = - (GMn)/R`
`E_p ∝ 1/R`
The graph for PE versus orbital radius R is shown 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.
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