Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 316 / 4 / iv / Solution

In the planetocentric frame, the incoming relative velocity ${\mathbf{w}}_i$ is prograde and the encounter outside the planet bends it inward through $\theta$, without changing $w=|\mathbf{w}|$. In the inertial frame $\mathbf{v}={\mathbf{v}}_p+\mathbf{w}$, soSubstitution of the previous result yields the negative gravitational assist energy kick