First slide

Motion of planets and satellites

Question

Two identical satellites are at R and 7R away from earth surface, the wrong statement is (R = Radius of earth)

Moderate

A

Ratio of total energy will be 4
B

Ratio of kinetic energies will be 4
C

Ratio of potential energies will be 4
D

Ratio of total energy will be 4 but ratio of potential and kinetic energies will be 2

Solution

Orbital radius of satellites $r_{1} = R + R = 2 R$
$r_{2} = R + 7 R = 8 R$
$U_{1} = \frac{- G M m}{r_{1}} a n d U_{2} = \frac{- G M m}{r_{2}} a n d \frac{U_{1}}{U_{2}} = \frac{r_{2}}{r_{1}} = \frac{8 R}{2 R} = 4 (∵ U \propto \frac{1}{r}) K_{1} = \frac{G M m}{2 r_{1}} a n d K_{2} = \frac{G M m}{2 r_{2}} a n d \frac{K_{1}}{K_{2}} = \frac{r_{2}}{r_{1}} = \frac{8 R}{2 R} = 4 (∵ K \propto \frac{1}{r}) E_{1} = \frac{G M m}{2 r_{1}} a n d E_{2} = \frac{G M m}{2 r_{2}} a n d \frac{E_{1}}{E_{2}} = \frac{r_{2}}{r_{1}} = \frac{8 R}{2 R} = 4 (∵ E \propto \frac{1}{r}) ∴ \frac{U_{1}}{U_{2}} = \frac{K_{1}}{K_{2}} = \frac{E_{1}}{E_{2}} = 4$

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