Planets and Satellites: Kepler's Laws — Question 3

Question 3

Conserved angular momentum fixes the speed ratio at the apsides

A satellite has periapsis distance $2R$ and apoapsis distance $5R$. Its periapsis speed is $v_p$. No noncentral force acts.

1. Find its apoapsis speed from angular momentum conservation.

2. Explain why the same formula $rv=\text{constant}$ cannot generally use total speed at an arbitrary intermediate point.

Question 3 - Solution

Conserved angular momentum fixes the speed ratio at the apsides

(a) Derivation and calculation

At an apsis, $\dot r=0$, so the entire velocity is transverse. Conservation of angular momentum gives $$mr_pv_p=mr_av_a,\qquad (2R)v_p=(5R)v_a.$$ $$\boxed{v_a=\frac25v_p}.$$

(b) Result and interpretation

At an arbitrary point, decompose velocity: $$\vec v=\dot r\hat r+r\dot\theta\hat\theta, \qquad L=mr(r\dot\theta)=mr v_t.$$ Therefore $$\boxed{rv_t=L/m=\text{constant}},\qquad v^2=\dot r^2+v_t^2.$$ Using $rv$ would incorrectly include radial motion in angular momentum.

Physical check

The greater apoapsis distance corresponds to the lower apsis speed, with equal products $r_pv_p=r_av_a$.