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7.Gravitation
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A planet orbits in an elliptical path of eccentricity $e$ around a massive star considered fixed at one of the foci. The point in space, where it is closest to the star is denoted by $P$ and the point, where it is farthest is denoted by $A$. Let $v_P$ and $v_A$ be the respective speeds at $P$ and $A$, then
A
$\frac{v_P}{v_A}=\frac{1+e}{1-e}$
B
$\frac{v_P}{v_A}=1$
C
$\frac{v_P}{v_A}=\frac{1+e^2}{1-e}$
D
$\frac{v_P}{v_A}=\frac{1+e^2}{1-e^2}$
(KVPY-2012)
Solution
(a)
By conservation of angular momentum,
We have,
$L_P=L_A$
$\Rightarrow \quad m v_P r_P=m v_a r_a$
So, ratio of speeds at points $P$ and $A$ is
$\frac{ V _{ P }}{ V _{ A }}=\frac{ r _{ A }}{ r _{ P }}=\frac{ a + ae }{ a – ae }$
$=\frac{a(1+e)}{a(1-e)}$
$=\frac{1+e}{1-e}$
Standard 11
Physics
