Double Ball Drop

If a light ball like a ping-pong ball is dropped along with a heavy ball like a large superball, the small ball rebounds with a remarkably high velocity, theoretically approaching three times the velocity with which the balls strike the surface. The analysis involves the nature of head-on elastic collisions and in particular the case of a light projectile hitting a heavy target. Slingshot orbits used in space exploration have features in common with this situation even though the objects involved never touch each other.

The rebound velocity of 3v for the small ball implies that its kinetic energy is nine times its incoming kinetic energy since the kinetic energy is proportional to the square of the velocity. Since the gravitational potential energy is proportional to the height and the kinetic energy is all converted to potential energy at the peak of the motion, it will rise to height 9h.
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Analysis of Double Ball Drop

The two slightly separated balls dropped from the same height are seen by a ground observer to approach the surface with velocity v.
A ground observer sees the larger ball hit and bounce up with velocity v while the smaller one still approaches. An observer on the larger ball would see the smaller one approach with velocity 2v. That observer would see the surface receding with velocity v.
Assuming perfectly elastic collisions and that the large ball is much more massive than the small one, the observer on the large ball will see the small one bounce back with velocity 2v. A ground observer would see the velocity of the small ball as 3v.

Double ball drop Standard collision examples Headon collision, massive target case
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Gravity-Assist or Slingshot Orbit

Planetary exploration vessels like Voyager I and Voyager II made use of propulsion maneuvers which gained energy from the planets and moons they passed. Such maneuvers are really elastic collisions where the objects involved never hit each other but are affected by gravity as they approach. An extreme case of such a propulsion maneuver is like the double ball drop.

A famous example of the use of such maneuvers is the exploration of Comet G-Z. The Voyager missions also made several energy-boosting flybys of planets on their paths through the solar system. The spacecraft Mariner 10 made several slingshot maneuvers in its exploration of Venus and Mercury.
More "gravity assist" maneuvers
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More Gravity-Assists

The use of a gravity-assist or "slingshot" orbit has been of great benefit in the exploration of the solar system. The spacecraft Ulysses used a gravity-assist from Jupiter to swing it far out of the ecliptic plane for observation of the sun's polar regions. Galileo used a gravity-assists from Venus, two flybys of Earth, and close encounters with asteroids Gaspra and Ida on its circuitous path to Jupiter. The more recent Cassini spacecraft used two gravity assist maneuvers from Venus and one from Earth before heading out to Jupiter and Saturn. The Dawn spacecraft used a gravity assist from Mars on its way to Vesta and Ceres. The New Horizons spacecraft used a gravity assist from Jupiter on its journey to Pluto.

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Collision concepts

Elastic collisions
 
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