In astronautical mechanics, the gravitational slingshot maneuver, which NASA calls a "gravity assist", exploits the gravitational attraction of a planet to alter the speed and trajectory of an interplanetary spacecraft. A spacecraft can thereby be accelerated by a near planetary flyby to enable considerable savings of fuel in missions to the outer planets, such as Jupiter and Saturn. At first sight, this might seem like a cosmic something-for-nothing scam. But the physics depends straightforwardly on conservation of momentum and energy and the huge planet-to-spacecraft mass ratio, which leaves the planetary orbit essentially undisturbed.

This Demonstration considers a hypothetical slingshot maneuver around the planet Jupiter (orange sphere with radius ≈ 143,000 km), which moves at an average speed of 13.1 km/sec in its orbit around the Sun. A spacecraft, with initial speed , which has a negligible mass and size compared to the planet, follows a hyperbolic path in Jupiter's frame of reference. In the Sun's frame of reference, however, the hyperbolic path is tilted and moves with velocity , which provides a terrific boost to the spacecraft after it crosses the orbit of the planet. The graphic shown is highly schematic, with both space and time scales significantly distorted. Refer to the references for more accurate formulas when you plan your next space mission.

The encounter of the spacecraft with the planet can be simulated by an elastic collision. In the simplest case of a head-on collision, the initial state can be represented by the diagram: ¤ and the final state by: ¤. The planet is so massive compared to the spacecraft that its motion is essentially unperturbed. In the actual situation, the final speed of the spacecraft is given by , where is the initial angle between and .