A planet of mass m moves in an elliptical orbit around the Sun of mass M . Let r 1 be the distance at aphelion and r 2 be the distance at perihelion. If v 1 and v 2 are the velocities of the p

Question

A planet of mass m moves in an elliptical orbit around the Sun of mass M . Let r 1 ​ be the distance at aphelion and r 2 ​ be the distance at perihelion. If v 1 ​ and v 2 ​ are the velocities of the planet at these points, then ratio of kinetic energy at aphelion to kinetic energy at perihelion is

TestHub · Accepted Answer

By conservation of angular momentum about the Sun, the angular momentum at aphelion equals the angular momentum at perihelion.

$m v_{1} r_{1} = m v_{2} r_{2}$

$\frac{v _{1}}{v _{2}} = \frac{r _{2}}{r _{1}}$

The kinetic energy of the planet is given by $K = \frac{1}{2} m v^{2}$ .

The ratio of kinetic energy at aphelion to kinetic energy at perihelion is

$\frac{K _{1}}{K _{2}} = \frac{\frac{1}{2} m v _{1}^{2}}{\frac{1}{2} m v _{2}^{2}} = (\frac{v _{1}}{v _{2}})^{2}$

Substituting the ratio of velocities, we get

$\frac{K _{1}}{K _{2}} = (\frac{r _{2}}{r _{1}})^{2} = \frac{r _{2}^{2}}{r _{1}^{2}}$

Answer: $\frac{r _{2}^{2}}{r _{1}^{2}}$

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