Let the speed of the planet at the perihelion Pin Figure be $v_{p}$ and the Sun-planet distance $SP$ be $r_{ P }$ Relate $\left\{r_{P}, v_{P}\right\}$ to the corresponding quantities at the aphelion $\left\{r_{A}, v_{A}\right\} .$ Will the planet take equal times to traverse $B A C$ and $C P B ?$
Let the speed of the planet at the perihelion Pin Figure be $v_{p}$ and the Sun-planet distance $SP$ be $r_{ P }$ Relate $\left\{r_{P}, v_{P}\right\}$ to the corresponding quantities at the aphelion $\left\{r_{A}, v_{A}\right\} .$ Will the planet take equal times to traverse $B A C$ and $C P B ?$
Answer The magnitude of the angular momentum at $P$ is $L_{p}=m_{p} r_{p} v_{p},$ since inspection tells us that $r_{p}$ and $v_{p}$ are mutually perpendicular. Similarly, $L_{A}=m_{p} r_{A} v_{A} .$ From angular momentum conservation
$m_{p} r_{p} v_{p}=m_{p} r_{A} v_{A}$
$\frac{v_{p}}{v_{A}}=\frac{r_{A}}{r_{p}}$
since $r_{A}>r_{p}, v_{p}>v_{A}$
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