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 elliptical orbit is given by

• A. $\frac{v_{p}}{v_{a}}$
• B. $\frac{v_{a}-v_{p}}{v_{a} + v_{p}}$
• C. $\frac{v_{p}+v_{a}}{v_{p} - v_{a}}$
• D. $\frac{v_{p}-v_{a}}{v_{p} + v_{a}}$

Answer 1

Answer: (D)

As $= \frac{v_{p}}{v_{a}} = \frac{a\left(1+e\right)}{a\left(1-e\right)}$
where a is semi-major axis of the ellipse
$v_{p}\left(1 - e\right) = v_{a} \left(1+e\right) $
or $v_{p} - ev_{p} = v_{a} + ev_{a}$
$e\left(v_{a}+v_{p}\right) = v_{p}-v_{a} $
or $e = \frac{v_{p} - v_{a}}{v_{a}+v_{p}}$