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December 1998
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Pat Murphy & Paul Doherty
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by Pat Murphy & Paul Doherty


Some of the best fantasy tales begin somewhere perfectly ordinary, take a left turn, and end up somewhere fantastic. You begin in Kansas and end up over the rainbow in Oz. You step into an old wardrobe and find yourself in Narnia.

Science sometimes works the same way. In this column, we're going to start with an ordinary baseball and an ordinary basketball, and we're going to end up on a grand tour of the solar system.

Trust us. It all connects up in the end.


You know about bouncing balls, right? You drop a tennis ball and it bounces. Give a moment's thought to how high it bounces. It bounces fairly high if it's just out of the pressurized can. It doesn't bounce much if it's been lying around and the dog's been chewing on it. But even if it's fresh out of the can--hey, even if you're using a super bouncy super ball--it never bounces higher than you were holding it when you dropped it, right?

When you drop the ball, gravity pulls it down and it picks up speed. It hits the ground and squashes at the moment of impact. As the squashed ball springs back to its original shape, it pushes on the floor and the floor pushes back. The force of the floor pushing against the ball throws the ball back up into the air.

To follow the bounce of the ball, scientists keep track of its energy. Energy, as you probably know, can't be created or destroyed. It can change from one form to another, but there's always the same amount of it around. When you lift the ball above the floor, you give it a certain amount of potential energy. When you drop the ball, that potential energy becomes kinetic energy, the energy of motion. That kinetic energy becomes the energy that deforms the ball. During the squashing some of the ball's energy dissipates as heat (bringing us that much closer to the Heat Death of the Universe, a subject with which Pat is obsessed and which will probably be the topic of a future column). The rest of the energy goes back into motion, carrying the ball back into the air. But because some of the ball's energy was lost as heat, the ball doesn't bounce as high as its starting point.

Knowing all this, you might figure that a ball could never bounce back higher than the height from which you dropped it. Right? Ah, don't agree too fast. If you've been reading this column for a while, you know that the world is sometimes tricky and things aren't always what they seem.


At the Exploratorium, we believe in experimenting. About a year ago, we were working with authors Susan Davis and Sally Stephens on The Sporting Life, a book about the science of sports. In the name of research, one day we took a variety of balls into the cavernous interior of the Exploratorium.

Standing amid an array of basketballs, tennis balls, golf balls, table tennis balls and baseballs, Paul held a tennis ball a meter above the ground and dropped it, watching how high it bounced. He repeated the experiment with other balls, one by one. All of the balls bounced back up to a height of less than three-quarters of a meter, always lower than their starting point. A quiet crowd gathered to watch the experiments.

Then Paul held a tennis ball on top of a basketball and dropped the two balls together. The tennis ball took off like a rocket, shooting over Paul's head. The crowd gasped. Paul grinned and took a bow. The crowd came to life and made us repeat the experiment again and again, suggesting other ball combinations.

Try the experiment for yourself! (Don't trust us. Surely you know better than that by now.) Get a larger, more massive bouncy ball and a smaller lighter ball. Some combinations that work well are a tennis ball and a basketball or a table tennis ball and a golf ball. Hold the more massive ball under the light ball and drop them at the same time.

Really. Try it. You'll be amazed. That feeling of amazement you get when you see an unexpected results is one of the great joys of science and should not be missed.

So what's going on here? We'll get to that in a minute. First, let's take a visit to outer space.


Back in the early 1980's NASA had a problem. They had been planning to launch the Galileo spacecraft to Jupiter. In NASA's original plan, the spacecraft was to be propelled directly toward Jupiter by a powerful Centaur rocket. That rocket was to have been carried into orbit by the space shuttle.

It was a fine plan--until the Challenger disaster in January of 1986. After that explosion, NASA reexamined the safety of carrying a liquid-fueled rocket inside the shuttle and decided that the risk was not acceptable. They began to investigate other alternatives.

They could launch the Galileo using the weaker, safer solid-fueled IUS (Inertial Upper Stage). But the IUS could not provide enough thrust to overcome the pull of the sun's gravity and propel the spacecraft to Jupiter.

What a dilemma! To solve it, NASA drew on the ideas of a mathematics graduate student at the University of California at Los Angeles: Michael Minovitch. We don't know if Minovitch ever dropped basketballs and tennis balls together, but his ideas relied on some of the same principles.

To get a feel for Minovitch's idea, think of the Galileo spacecraft as a table tennis ball and the planet of Venus as a basketball. What NASA did, in essence, is bounce the Galileo spacecraft off of Venus to give it a velocity boost. Then NASA bounced the spacecraft off of the earth twice to send it shooting toward Jupiter, a maneuver known as a "gravity assist."

How do you bounce a spacecraft off a planet? Let's take a closer look at how a gravity assist works. If you only look at half of the picture, it all seems pretty straightforward. If you drop a spacecraft into a planet, the gravitational attraction of the planet speeds it up, just like a dropped ball speeds up on its way to the floor.

Let's assume that you've very carefully aimed your spacecraft so that it doesn't actually hit the planet. Instead, it executes a near-miss. (An interesting aside here: that's basically what an orbit is. A near miss that goes on and on and on. A spacecraft in orbit is always falling toward the planet, but always missing.)

The spacecraft misses the planet, but doesn't get to keep that speed it gained. After the spacecraft races past the planet, gravity slows it down on the way out, just as the ball slowed down on its way up. In the end the spacecraft leaves the planet with the same speed with which it arrived.

How can we set this up so that passing the planet gives the spacecraft a boost? To figure out what's going on, let's take another look at the bouncing balls and why they act as they do.


Consider a tennis ball riding piggyback on a basketball, starting from rest in Paul's hands at 1 meter off the floor. The balls accelerate toward the floor and are going about 4 meters/second when they hit. The basketball hits the floor first and reverses direction, heading up at 4 meters/second. The tennis ball is still going down at 4 meters/second.

At least, that's how fast the tennis ball is going if you're watching it from the point of view of someone standing on the ground. But suppose you were a tiny person standing on the surface of the basketball? (Or, as Paul would prefer to put it, suppose you were watching from the frame of reference of the basketball.) From the point of view of someone standing on the floor, the basket ball is traveling up at 4 meters/second and the tennis ball is traveling down at 4 meters/second. But from the surface of the basketball, you'd see the tennis ball traveling toward you at 8 meters/second. Its speed relative to you would be 8 meters/second.

You can compare this shift in viewpoint to driving down the road at 60 mph. On the other side of the double yellow line, a car is coming toward you at 60 mph. Relative to the road, you're traveling 60 mph and the other car is traveling 60 mph. But your speed relative to that car is 120 mph.

So the tennis ball smacks into the basketball and heads in the other direction. Since little energy is lost in the collision, the tennis ball leaves the basketball at nearly the same speed at which it arrived. Since the basketball is more massive than the tennis ball, the collision doesn't slow down the basketball much. The basketball slows down only a little, but the tennis ball reverses direction. From your viewpoint on the basketball, the relative speed of the balls remains constant. After the balls hit, they separate at 8 meters/second.

Ah, but here's the tricky bit. For a person standing on the ground and watching the balls bounce, the picture is different. That basketball is still moving up at 4 meters/second. The tennis ball is going up 8 meters/second faster than the basketball. So the tennis ball is moving up at 12 meters/second, rather than just 4 meters/second. That's triple its original speed with respect to the earth! With triple the speed, the ball bounces 9 times higher than the height from which it was dropped, shooting over Paul's head and amazing the spectators.

Where did it get the energy to do this? From the basketball. It takes a lot of energy to move that massive basketball. When the tennis ball bounced off the basketball, it gained just a little bit of the basketball's kinetic energy. If you watched really closely, you'd notice that the basketball dropped in tandem with the tennis ball doesn't bounce quite as high as the basketball dropped alone. That's because the tennis ball stole a bit of the basketball's energy.

The general rule is easy: when a ball bounces off a much heavier moving object and doesn't lose any energy to heat, it reverses its direction and gains twice the speed of the object it bounced off of. This means that a baseball leaves the batter at the speed the pitcher threw the ball plus twice the speed of the bat (minus some speed lost as a result of heat). It also means that a golf ball that is initially at rest leaves the tee at twice the speed of the striking club head (again minus a bit for heat).


To understand how a gravity assist works all you need to do is be able to add and subtract and imagine yourself in different places (that is, different frames of reference). The Galileo spacecraft made a gravity assist flyby of Venus and then returned to earth for two more gravity assists. We'll consider one of the earth flybys.

Let's say you've got a spacecraft that's orbiting the sun at the same distance as the earth. The spacecraft is traveling in the opposite direction as the earth--the earth orbits counterclockwise, and the spacecraft orbits clockwise. Both are going 30 kilometers/second. The spacecraft comes in towards the planet, swings around it in a cosmic do-se-do, and leaves moving out along the line of its approach.

That's your planetary collision. You may wonder why we call this a collision, since the spacecraft didn't touch the planet. Pat expresses her sympathy with this sensible viewpoint, but defers to the physicists. Paul and his fellow physicists consider this a collision even though the spacecraft doesn't touch the earth. When pressed, Paul says he draws a large sphere around the earth and watches the spacecraft enter and leave the sphere. If the spacecraft changes its direction or speed while inside the sphere he knows it has suffered a collision. He says that the spacecraft interacts with the planet via the long range force of gravity, not the short range electric forces that come into play when two objects actually touch.

And Pat concedes that Paul has a point. The spacecraft does act just like it has collided with something. The relative speeds of the two objects don't change. The spacecraft and planet come together at 60 kilometers/second and leave each other at 60 kilometers/second. That's what you see if you are standing on the earth, which is the equivalent of the basketball in this situation.

But suppose you back up and look at the collision in the frame of the distant stars. Then you see a spacecraft initially orbiting the sun at 30 kilometers/second. After the collision, you see a spacecraft going 90 kilometers/second! The spacecraft is leaving the earth at 60 kilometers/second and the earth is going 30 kilometers/second so 60 + 30 = 90! That's fast enough to give the spacecraft escape velocity from the sun , heading out toward interstellar space along a hyperbolic trajectory

The spacecraft gains kinetic energy in this encounter. Where does that energy come from? Well, just as the encounter with the tennis ball slowed the basketball down, the encounter with Galileo slowed the earth down. Not by much, of course. When the Galileo spacecraft swung by earth, it sped up by over 16,000 kilometers per hour with respect to the sun, and the earth slowed down by 10 billionths of a centimeter per year. A reasonable trade, we figure.


Michael Minovitch's calculations showed NASA that the outer planets were lined up so that they could be used to give a spacecraft multiple gravity assists to allow that spacecraft to make a grand tour of the solar system. Voyager 2 used a gravity assist from Jupiter to propel it to Saturn, then used Saturn to get to Uranus and finally used Uranus to get to Neptune and beyond. Voyager is now continuing on its way out of the solar system. In all of its gravity assists, it gained escape velocity and will never return to the sun.

We haven't figured out how to do that with a tennis ball just yet. Paul has calculated that it is theoretically possible for a ping pong ball to achieve escape velocity if you balance it on a stack of 9 other balls, each much more massive than the one above it, each bouncier than a superball. Drop the balls 5 meters, or one story, under these ideal circumstances, and the ping pong ball would end up traveling at 11 kilometers/second, fast enough to escape the earth's gravitational field. We haven't managed that yet, but we're still experimenting.

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