13.7 Exercise 26

Given the perihelion distance, p, and aphelion distance, q, for an elliptical orbit, show that the velocity at perihelion, $v_{p}$, is given by $v_{p} = \sqrt{\frac{2 G M_{\text{Sun}}}{\left(q + p\right)} \frac{q}{p}}$. (Hint: Use conservation of angular momentum to relate $v_{p}$ and $v_{q}$, and then substitute into the conservation of energy equation.)