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The ratio of velocities of a planet at perigee and apogee $\frac{v_p}{v_a}$ is equal to -
Option: 1
1

Option: 2
$\frac{1-e}{1+e}$

Option: 3
$\frac{1+e}{1-e}$

Option: 4
$\frac{1+e^2}{1-e^2}$

Answers (1)

best_answer

As we have learnt,

Velocity of planet in terms of Eccentricity -

$V_{a}=\sqrt{\frac{GM}{a}\left ( \frac{1-e}{1+e} \right )}$

$V_{p}=\sqrt{\frac{GM}{a}\left ( \frac{1+e}{1-e} \right )}$

$V_{A}=$ Velocity of planet at apogee

$V_{p}= Velocity of perigee$

- wherein

Eccentricity (e) = $\frac{c}{a}$

$r_{p}=a-c$

$r_{a}=a+c$

By conservation of angular momentum,

$mv_pr_p = mv_ar_a$

$\Rightarrow \frac{v_p}{v_a} = \frac{r_a}{r_p} = \frac{a+c}{a-c} = \frac{1+e}{1-e}$

Posted by

manish

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