Question

A planet revolves around the sun in an elliptical orbit. If $ {V_p} $ and $ {V_a} $ are the velocities of the planet at the perigee and apogee respectively, then the eccentricity of the elliptical orbit is given by:
(A) $ \dfrac{{{V_p}}}{{{V_a}}} $
(B) $ \dfrac{{{V_a} - {V_p}}}{{{V_a} + {V_p}}} $
(C) $ \dfrac{{{V_p} + {V_a}}}{{{V_p} - {V_a}}} $
(D) $ \dfrac{{{V_p} - {V_a}}}{{{V_p} + {V_a}}} $

Answer 1

Hint: Planets revolve around the sun in elliptical orbits. Perigee and apogee are two extreme points of the elliptical orbit. Perigee is the point on the ellipse when the planet is closest to the sun and apogee is the point where the planet is the farthest from the sun.

Formula used:
Perigee is the point where the planet is closer to the sun. The velocity of the planet at perigee can be written as,
 $ {V_p} = a(1 - e) $
Apogee is the point where the planet is the farthest to the sun. The velocity of the planet at the apogee can be written as
 $ {V_n} = a(1 + e) $
where $ {V_p} $ is the velocity at the perigee, $ {V_a} $ stands for the velocity at the apogee, where $ a $ is the semi-major axis, $ e $ stands for the eccentricity of the ellipse.

Complete step by step solution
The ratio of the velocities can be written as,
 $ \dfrac{{{V_p}}}{{{V_a}}} = \dfrac{{a(1 - e)}}{{a(1 + e)}} $
Canceling the common terms, we get
 $ \dfrac{{{V_p}}}{{{V_a}}} = \dfrac{{(1 - e)}}{{(1 + e)}} $
Cross multiplying, we get
 $ {V_p}(1 + e) = {V_a}(1 - e) $
Opening the brackets,
 $ {V_p} + e{V_p} = {V_a} - e{V_a} $
Taking the terms containing $ e $ on one side,
 $ e({V_p} + {V_a}) = {V_a} - {V_p} $
From this, we can write the eccentricity as,
 $ e = \dfrac{{{V_a} - {V_p}}}{{{V_p} + {V_a}}} $
The answer is: Option (B); $ \dfrac{{{V_a} - {V_p}}}{{{V_a} + {V_p}}} $ .

Additional information
The furthest any body can get from the sun is called the aphelion. The generic term is called apoapsis. It is the furthest distance any body can get from another body in an elliptical orbit. The nearest any body can get from the sun is called perihelion. The generic term is periapsis.

Note
All planets and moons revolve around the sun in elliptical orbits. The semi-major and semi-minor axes of the ellipse are the two parameters of the elliptical orbit. The eccentricity describes the shape of the ellipse. If we know any two of apogee, perigee, eccentricity, and semi-major axis, we can easily calculate the other two.