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The speed of a palnet at apogee which revolves around the sun in an elliptical orbit,in terms of eccentricity is

The speed of a palnet at apogee which revolves around the sun in an elliptical orbit,in terms of eccentricity is

Grade:12

3 Answers

Saurabh Koranglekar
askIITians Faculty 10335 Points
4 years ago
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Vikas TU
14149 Points
4 years ago
Arun
25750 Points
4 years ago
Apogee is the point in the elliptical orbit where the planet is farthest from the sun and perigee is the point where the planet is nearest to the sun. To calculate the ratio of K.E , we use the conservation of angular momentum.
So,
Angular momentum at apogee = Angular momentum at perigee
Let,
Distance at apogee = r₍a₎
Distance at perigee = r₍p₎
The values of r₍a₎ and r₍p₎ can be represented in terms of semi major axis and eccentricity e. If 'a' is the length of semi major axis then,
r₍a₎ = a (1+e)
r₍p₎ = a (1-e)
Now, by conservation of angular momentum,
m v₍a₎ r₍a₎ = m v₍p₎ r₍p₎   .......... (1)
where m is the mass and v₍a₎ and v₍p₎ are the velocities of apogee and perigee respectively.
Equation (1) ⇒ v₍a₎ / v₍p₎ = r₍p₎ / r₍a₎
v₍a₎ / v₍p₎ = = (1-e) / (1+e)
The ratio of K.E is given as,
(1/2mv₍p₎²) / (1/2mv₍a₎²) = ( v₍a₎ / v₍p₎ )²
(1/2mv₍p₎²) / (1/2mv₍a₎²) = [(1+e) / (1-e)]²

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