The gravitational slingshot effect: Figure shows the planet Saturn moving in the negative x direction at its orbital speed (with respect to the Sun) of 9.6km/s. The mass of Saturn is 5.69×1026kg. A spacecraft with mass 825 kg approaches Saturn. When far from Saturn, it moves in the +x direction at 10.4 km/s. The gravitational attraction of Saturn (a Conservative force) acting on the spacecraft causes it to swing around the planet (orbit shown as dashed line) and head off in the opposite direction. Estimate the final speed (in km/s) of the spacecraft after it is far enough away to be considered free of Saturn’s gravitational pull.

Question

The gravitational slingshot effect: Figure shows the planet Saturn moving in the negative x direction at its orbital speed (with respect to the Sun) of 9.6km/s. The mass of Saturn is  5.69×1026kg. A spacecraft with mass 825 kg approaches Saturn. When far from Saturn, it moves in the +x direction at 10.4 km/s. The gravitational attraction of Saturn (a Conservative force) acting on the spacecraft causes it to swing around the planet (orbit shown as dashed line) and head off in the opposite direction. Estimate the final speed (in km/s) of the spacecraft after it is far enough away to be considered free of Saturn’s gravitational pull.

Accepted Answer

The interaction between the planet and the spacecraft is elastic, because the force of gravity is conservative. Thus kinetic energy is conserved in the interaction. Consider the problem a 1-dimensional collision, with A representing the spacecraft and B representing Saturn. Because the mass of Saturn is so much bigger than the mass of the spacecraft, Saturn’s speed is not changed appreciably during the interaction. Use Eq., with $v_{A} = 10.4 k m / s$ and $v_{B} = v'_{B} = - 9.6 k m / s$ .

$v_{A} - v_{B} = - v'_{A} + v'_{B} \to v'_{A} = 2 v_{B} - v_{A} = 2 (- 9.6 k m / s) - 10.4 k m / s = - 29.6 k m / s$

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