Question

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

Answer 1

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

Answer 2

Answer:

(1-e)^2/(1+e)^2

Explanation: