Consider an elliptical orbit of a planet around the Sun with perihelion (nearest)…

PHYSICS · Subjective · AP · Physics 1: Algebra-based · Energy and Momentum of Rotating Systems · Energy and Momentum of Rotating Systems

Problem

Consider an elliptical orbit of a planet around the Sun with perihelion (nearest) distance

r_p

and aphelion (farthest) distance

r_a

. (a) Write the expression for the angular momentum of the planet about the Sun when it is at perihelion and when it is at aphelion, taking its speeds to be

v_p

and

v_a

respectively. (b) Hence derive a relation connecting

v_p,\,v_a,\,r_p

and

r_a

. (c) Show that the ratio of the perihelion speed to the aphelion speed is equal to the ratio of the aphelion distance to the perihelion distance, that is,

\frac{v_p}{v_a}=\frac{r_a}{r_p}.

(d) If a comet has

r_p = 0.5\,\text{AU}

and

r_a = 10\,\text{AU}

, find the factor by which its speed changes between perihelion and aphelion.

Related theory

Conservation of Angular Momentum
Angular Momentum

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