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Q.

A planet moves around the sun of mass m in an elliptical orbit of eccentricity e. If a is the semi major axis of the elliptical orbit, the velocity of the planet at the apogee is given by....

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a

$\sqrt{\frac{Gm}{a} [\frac{1 + e}{1 - e}]}$

b

$\sqrt{\frac{Gm}{a} [\frac{1 - e}{1 + e}]}$

c

$\sqrt{\frac{2 Gm}{a} [\frac{1 - e}{1 + e}]}$

d

$\sqrt{\frac{2 Gm}{a} [\frac{1 + e}{1 - e}]}$

answer is B.

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Detailed Solution

By Law of C.A.M

Question Image

$m r_{a} V_{a} = m r_{p} \cdot V_{p}$

$\Rightarrow a (1 + e) V_{a} = a (1 - e) V_{p}$

$\Rightarrow V_{a} (1 + e) = V_{p} (1 - e)$ .......(1)

From L.C.E

$\frac{- GMm}{a (1 + e)} + \frac{1}{2} {mV}_{a}^{2} = \frac{- GMm}{a (1 - e)} + \frac{1}{2} {mV}_{p}^{2}$ ......... (2)

Solving (1) & (2) $\Rightarrow V_{a} = \sqrt{\frac{GM}{a} [\frac{1 - e}{1 + e}]}$

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A planet moves around the sun of mass m in an elliptical orbit of eccentricity e. If a is the semi major axis of the elliptical orbit, the velocity of the planet at the apogee is given by....