Gravity assist

Remarks

The size of the change is $\norm{u_{\mathrm{out}} - u_{\mathrm{in}}} = 2 v_\infty \sin(\delta/2)$. The heliocentric speed changes because $\norm{V + u_{\mathrm{out}}}$ differs from $\norm{V + u_{\mathrm{in}}}$ in general. It grows when the craft passes behind the planet, so that $u_{\mathrm{out}}$ turns towards $V$.

Example: let $u$ be parallel to $V$ with $\norm{u} = v_\infty$, and let $u_{\mathrm{in}} = -u$ and $u_{\mathrm{out}} = u$. Then the heliocentric speed goes from $\norm{V} - v_\infty$ to $\norm{V} + v_\infty$, a gain of $2 v_\infty$. This is the limit $\delta \to \pi$, which needs $r_p \to 0$.

Non-example: if the attracting body does not move, $V = 0$, then $\norm{v_{\mathrm{out}}} = \norm{v_{\mathrm{in}}}$, so the flyby only turns the craft and its speed relative to the body is unchanged.

The total energy of the craft and the planet is conserved; the planet's change of speed is negligible because of its large mass.