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# 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) $\dfrac{{{V_p} - {V_a}}}{{{V_p} + {V_a}}}$

Last updated date: 16th May 2024
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Hint ${V_p} = a\left( {1 + e} \right)$ and ${V_a} = a\left( {1 - e} \right)$ [where, $a$ is the semi-major axis of the ellipse and $e$ is the eccentricity]. Divide both the velocities and cross multiply both sides. Take the terms on the same sides.
Formula used: Let, $a$ be the semi-major axis of the ellipse and $e$ is the eccentricity of the elliptical orbit.
So, ${V_p} = a\left( {1 + e} \right)$ and ${V_a} = a\left( {1 - e} \right)$

Complete step by step solution:
Eccentricity of an ellipse is defined as the ratio of the distance between the center of the ellipse and each focus to the length of semi-major axis.
It is given that ${V_p}$ and ${V_a}$ are the velocities of the planet at the perigee and apogee respectively.
Now, let $a$ be the semi-major axis of the ellipse and $e$ is the eccentricity of the elliptical orbit.
Then, $\dfrac{{{V_p}}}{{{V_a}}} = \dfrac{{a\left( {1 + e} \right)}}{{a\left( {1 - e} \right)}}$
or, ${V_p}\left( {1 - e} \right) = {V_a}\left( {1 + e} \right)$ [cross multiplying both the sides]
or, ${V_p} - e{V_p} = {V_a} + e{V_a}$
or, $e\left( {{V_p} + {V_a}} \right) = \left( {{V_p} - {V_a}} \right)$ [taking the like terms on the same sides]
or, $e = \dfrac{{{V_p} - {V_a}}}{{{V_p} + {V_a}}}$

Note: It should be always remembered that the eccentricities of the orbits of the planets in the solar system maintain all Kepler’s laws. We can say that the orbits are almost circular. As the circle is a special case of an ellipse (although for circles $e = 0$ ), Kepler’s laws are equally applicable for the circular orbits too. So, we can say all Kepler’s laws are applicable and valid for both the circular and elliptical orbits, which means for orbits having $0 \leqslant e < 1$ .