Elastic Collisions

To simplify the physics, the ping-pong ball is assumed to be "perfectly elastic", losing no energy when it rebounds. If such balls existed and one of them was dropped onto a hard floor, it would not just bounce back, but would rise to the same height as the one it started from. Real ping-pong balls don´t quite match such perfection, but spacecraft in planetary fly-by maneuvers, like the ones discussed further below, do so.

    A general problem of this sort, with ball and paddle both moving at an angle, requires the use of vectors. But head-on collisions are simpler, for velocities can then be viewed as positive in one direction and negative in the opposite one. We then can imagine a ball traveling at -20 mph meeting a paddle moving at +20 mph (top figure; the rebound is drawn at an angle, though it is really in the opposite direction). What happens next?

    In any problem involving forces, adding the same velocity to all motions does not change the physics. Inside a jetliner moving at 600 mph dropped objects fall straight down, just as they do inside a house. The reason is, of course, that adding 600 mph to all motions leaves accelerations (changes in velocity) the same, and by Newton's laws, only accelerations require forces to act.

    We may therefore add -20 mph to all velocities. The paddle's velocity is now zero--which means we are moving with it or "are in its frame of reference"--while the ball now moves at (-20) - 20 = -40 mph (middle drawing). In the new problem, a ball moving at -40 mph hits a non-moving surface; if the collision is elastic (no energy lost), that ball returns with the same speed but in the opposite direction, i.e. its velocity reverses sign. Its rebound velocity is therefore +40 mph.

    Finally we return to the real world, by adding +20 mph to all velocities. The paddle now has again 0 + (20) = 20 mph, its original velocity, while the ball has 40 + 20 = 60 mph. Colliding with the moving paddle has increased the ball's speed v by twice the paddle's velocity, and since kinetic energy goes as v², the tripling of the speed from 20 to 60 mph means that energy has increased 9-fold! The extra energy was supplied by the paddle and by the hand holding it.

    This sort of maneuver is often used by interplanetary probes. Encounters with the Moon have also been used, for sending spacecraft towards the Lagrangian L1 point or the L2 point in the tail, for modifying missions (such as that of Geotail) and on December 22, 1983 for sending ISEE-3 away from Earth towards Comet Giacobini-Zinner (the figure above describes its orbit, which includes several encounters with the Moon, on the dates given). Even encounters with the Earth itself have been used this way, e.g. by the space probe Galileo sent to orbit Jupiter: the probe first went to Venus, then came back to swing by Earth. All this looks like a cosmic game of billiards, except that the "cue ball," the satellite making the encounters, carries its own small rocket engine, to correct its course as needed and thus make complex shots possible.

    The first use of this maneuver was by Mariner 10 launched in 1973 towards Venus, which used that planet's gravity to extend its own orbit to the planet Mercury. The probe Ulysses used an encounter with Jupiter in order to fling itself out of the ecliptic, into an orbit passing above both poles of the Sun. And NASA's "solar probe ," intended to come within 4 solar radii of the center of the Sun, may use a tight "hairpin" trajectory around Jupiter to reverse most of its outward velocity, allowing it to make a close approach of the Sun

  One vocal proponent of using gravity assist maneuvers for economical flights between Earth and Mars has been the astronaut Buz Aldrin. His ideas are described in the article "A Bus Between the Planets" by James Oberg and Buzz Aldrin, p. 58-60, "Scientific American," March 2000.

  An interesting parallel exists between Planetary swing-by maneuvers and the operation of the Pelton water turbine. More on this in section #35a, Planetary Gravity-Assist and the Pelton Turbine.

To the Stars

    Close encounters with attracting planets have another use: the energy added by rocket thrust is greatest when applied at closest approach. A proposed "Profile" mission to the Earth's magnetosphere calls for a dozen small satellites placed initially aboard a spinning "mother ship" in a long elliptical orbit. At each closest approach ("perigee") two satellites are let go, letting the rotation fling one forwards and the other backwards. With no more extra speed than that of a runner in a footrace, that boost is enough, by the next closest approach, to place them on two neighboring orbits with periods differing by one hour. At the next perigee pass another pair of satellites is let go, and so on, two at each perigee. In the end all satellites may be strung out one hour apart in two new orbits (drawing on right), which differ slightly from the orbit of the mother ship. The two groups also overtake each other, and each group clusters together near apogee, the greatest distance from Earth. A rather gentle push does it all, but it must be delivered at the right place.

    Some day in the far future humanity will probably want to send robot explorers to the distant stars, journeys that may require many thousands of years. Long ago it was proposed that a good way to achieve the required speed would be to first pass close to the Sun, and at the closest approach fire a rocket. Even a modest increase in speed near the Sun then translates to a great increase in the ultimate velocity far away.

    Unfortunately, this requires getting rather close to the Sun, and the space probe is likely to melt. A more reasonable start for a mission to a star may be from a comet, because the trajectories of comets already extend to the edge of the solar system, and the ice found on them may provide hydrogen and oxygen for fueling rocket motors.

    In this manner the stars nearest to Earth might be reached. To reach more distant parts of the galaxy, however, may require close encounters like those proposed with the Sun. Dwarf stars exist which are as compact as Earth and as massive as the Sun, and perhaps some can be located that are burned out, dark and cold. If so, they would be a perfect target for such a maneuver. All that is now just pure fantasy: but humanity has the time, its venture into space is just beginning.

    Added note: According to recent surveys, dim dwarf stars are surprisingly abundant in the Sun's neighborhood, at least 60% of the total count. See Science, 11 June 2004, p. 1587.

Exploring Further:

"The Starflight Handbook" (subtitle: "A Pioneer's Guide to Interstellar Travel"), by Eugene F. Mallove and Gregory L. Matloff, John Wiley & Sons, 1989.

"Voyager 2" is currently on its way out of the solar system. On the off-chance that some day in the distant future, an alien civilization might find it, a message was sent with it. See here for details

Related file: #35a. Planetary Gravity-Assist and the Pelton Turbine

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Author and Curator:   Dr. David P. Stern
     Mail to Dr.Stern:   stargaze("at" symbol)phy6.org .


Updated 9-24-2004 later update 26 October 2016