= Gravity assist
A <gravity assist> is a <spacecraft> flyby that uses the <gravity> of a moving planet or other massive body to change the spacecraft's trajectory and its orbital <energy> relative to another body, such as the Sun. In the ideal unpowered two-body encounter, the incoming and outgoing speeds far from the planet are equal in the planet's <reference frame>. Their directions differ. Adding the planet's motion converts that directional change into a possible speed change in the Sun's <reference frame>.<Basics of Spaceflight: A Gravity Assist Primer by NASA>[planet-relative incoming/outgoing velocity vectors and planetary-motion vector]{ref}
Write the planet's nearly constant <velocity> during the encounter as $\mathbf V$, and the incoming and outgoing planet-relative velocities as $\mathbf u_i$ and $\mathbf u_o$. Then $\mathbf v_i=\mathbf V+\mathbf u_i$, $\mathbf v_o=\mathbf V+\mathbf u_o$, and $|\mathbf u_i|=|\mathbf u_o|$. Subtracting the squared speeds gives the local change in specific orbital <energy>:
$$
\Delta\epsilon=\frac{|\mathbf v_o|^2-|\mathbf v_i|^2}{2}=\mathbf V\cdot(\mathbf u_o-\mathbf u_i).
$$
For a hypothetical $90^{\circ}$ turn, choose $\mathbf V=(10,0)$, $\mathbf u_i=(-5,0)$ and $\mathbf u_o=(0,5)$, all in $\mathrm{km\,s^{-1}}$. Planet-relative speed stays $5$, while Sun-relative speed changes from $5$ to $\sqrt{125}\approx11.18\,\mathrm{km\,s^{-1}}$. The specific-energy gain is $50\,\mathrm{km^2\,s^{-2}}$. Whether a given turn is physically available depends on the planet's mass, the incoming speed and the permitted closest approach.
<Conservation of energy> is maintained because the planet and spacecraft exchange momentum and energy. The planet's change is extremely small because of its much greater mass. A different encounter geometry can remove orbital <energy> or mainly change <orbital inclination>; <Ulysses spacecraft> used the latter possibility to reach high solar latitudes.<Gravity Assists for Cassini by NASA>[three-body interpretation and conservation-of-energy discussion]{ref}
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