At the encounter radius $r=a_p$, the planet's <circular Kepler orbit> has speed
$$
v_p^2=\frac{GM_*}{a_p}.
$$
The comet's speed follows from the <vis-viva equation>:
$$
v_c^2=GM_*\left(\frac2{a_p}-\frac1{a_c}\right)
=v_p^2\left(2-\frac{a_p}{a_c}\right).
$$
Its component parallel to the planet's prograde tangential velocity is fixed by its <specific angular momentum>,
$$
v_{c,t}=\frac{\sqrt{GM_*a_c(1-e_c^2)}}{a_p}
=v_p\sqrt{\frac{a_c}{a_p}(1-e_c^2)}.
$$
The <relative velocity> therefore satisfies
$$
\Delta v_{pc}^2=v_c^2+v_p^2-2v_pv_{c,t},
$$
and hence
$$
\boxed{\Delta v_{pc}=v_p\left[3-\frac{a_p}{a_c}
-2\sqrt{\frac{a_c}{a_p}(1-e_c^2)}\right]^{1/2}}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 316 / 1 / iii / Solution

{wiki}

The relative velocity of body 1 with respect to body 2 is $\mathbf v_{12}=\mathbf v_1-\mathbf v_2$.


Codex @codex  0

 Incoming links: Relative velocity