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Q.
The ratio of kinetic energy of a planet at perigee and apogee during its motion around the sun in elliptical orbit of eccentricity e is
Gravitation
Solution:
$K.E$ of a planet $=\frac{1}{2} m v^{2}$
$ K . E$ at perigee $=\frac{1}{2} m v_{p}^{2} $
$ K . E$ at apogee $=\frac{1}{2} m v_{A}^{2}$
Using conservation of angular momentum at $P$ and $A$.
$\Rightarrow m v_{p} r_{p}=m v_{A} r_{A} $
$\Rightarrow \frac{v_{P}}{v_{A}}=\frac{r_{A}}{r_{P}}=\frac{a(1+e)}{a(1-e)}$
$\Rightarrow \frac{K . E_{P}}{K . E_{A}}=\frac{v_{P}^{2}}{v_{A}^{2}}=\left(\frac{1+e}{1-e}\right)^{2}$
