Past exam of the physics 1 paper of the University of Oxford / 2017 / 11 / e

Take Jupiter's heliocentric velocity as ${\mathbf{V}}_J=13\hat{\mathbf{x}}\mathrm{km}{\mathrm{s}}^{-1}$. Conservation of energy in the Jupiter frame makes the incoming and outgoing asymptotic speeds both $10\mathrm{km}{\mathrm{s}}^{-1}$. For the $90^{\circ }$ encounter, an outgoing velocity in Jupiter's direction requires the incoming Jupiter-frame velocity to be perpendicular to it:Use a Galilean transformation back to the Sun frame:The heliocentric speeds are thereforeThe incoming direction makes an angle $\arctan (10/13)\simeq 37.6^{\circ }$ with Jupiter's motion; its two signs describe mirror-image encounter planes. The gravitational assist transfers orbital energy from Jupiter to the spacecraft: the spacecraft's specific kinetic energy increases by $\frac{1}{2}(23^2-16.4^2)\mathrm{k}{\mathrm{m}}^2{\mathrm{s}}^{-2}$, or, using the unrounded squared incoming speed, exactly $130\mathrm{k}{\mathrm{m}}^2{\mathrm{s}}^{-2}=1.30\times 10^8\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}$. There is no gain in asymptotic speed in the Jupiter frame; the gain is a consequence of changing frames while the planet is moving.