So your mum won't let you set fire to anything in the garden? You don't have the spare change to spend on rocket fuel? You don't really understand thermodynamics and higher mathematics? You're ready to launch a rocket at the press of a button, but you're worried about the size of your carbon footprint? The police have politely requested that you stop handling explosives? Despair not! There might still be a way to make your personal, autographed bit of space junk join the hundreds of millions of pieces of man-made debris already floating around in our solar system. And don't worry - despite the fact that it involves bouncing balls, it's still perfectly geeky, and things will go boom.

To follow this entry, you will need a good imagination and some credulity - or, better yet, two bouncy balls of different sizes. Good combinations would be a basketball and a tennis ball or a tennis ball and a small superball.

Understanding Bounciness

Once upon a time, there was a little girl called Goldilocks. On a beautiful summer morning, she was out walking in the woods when she happened upon a most charming little clearing with a most charming little house sitting in it. Since she was a nosy little girl, she walked right through the gate and marched up the path to the front door, where she found three balls lined up on the windowsill - a soft, crocheted beanbag, a perfect, polished steel sphere, and a bright red rubber ball. She picked up the first ball. When she threw it down on the stoop, it just stayed there in a rather disappointing splat of yarn. 'Too soft!' thought Goldilocks and picked up the second ball. When she threw it down on the stoop, it skipped back up a few centimetres and rolled off into the underbrush. 'Too hard!' thought Goldilocks, and picked up the third ball. When she threw it down on the stoop, it bounced up high, landing in the gutter. 'Just right!' thought Goldilocks. Unfortunately, she was almost entirely wrong about this...

It's Not The Hardness, It's What You Do With It

So why does a ball bounce? To put it simply, because it's elastic. That means that when a force¹acts on the ball, the ball is deformed - compressed in the direction in which the force is acting - and when the force stops acting, the ball springs back into its original shape. Imagine squeezing a tennis ball² between your palms. When you press, the ball is flattened slightly. If you relax your muscles, the ball will become round again, slightly pushing your palms apart. The same thing happens when you drop the ball on the ground - the ball is deformed slightly, then pulls itself back into shape, pushing against the ground and launching itself upward.

It is impossible to lose energy³. It can merely be misplaced - that is, converted into another form of energy⁴. When you squeeze the ball, the kinetic energy of your moving muscles is converted into elastic potential energy, which is stored in the ball and makes it spring back into shape, exerting nearly the same force outward that you exerted inward - though a small amount of energy is converted into heat, and, to some degree, goes toward permanently deforming the ball.

There are two ways in which a ball can be made elastic enough to bounce properly. The first is to make it out of an elastic material, such as a rubber-like polymer, which springs back into shape when deformed because of electromagnetic forces acting on its atoms. The other is to construct a hollow sphere and pump it full of air - or any other gas - at more than the outside air pressure⁵. Since the gas molecules like their personal space, they will go as far away from every other gas molecule as possible, resulting in the gas being evenly distributed in the sphere - like sunbathers on a beach. When a force acts on the ball, deforming it, the gas molecules are forced to move closer together⁶, like the sunbathers when the tide comes in. When the pressure is removed, they spread out evenly again. Of course, it is important that the skin of the ball be made of an easily deformable material, such as a thin layer of plastics, because otherwise, more energy will go towards deforming it, leaving less for the bounce. Higher pressure inside generally means more bounce, as the air can 'push back harder'.

How High Can You Bounce?

When you drop a ball, no matter how well inflated, it won't bounce as high as it started. Some of this is due to air resistance slowing it down. Even if we ignore air resistance - something physicists like to do - the ball won't quite reach its starting point, due to the energy 'lost', that is, transformed into heat and other 'useless' forms of energy during the bounce. The degree to which a ball will bounce back in a vacuum, that is, with no air resistance, and when falling onto a perfectly firm surface that will not move or be deformed, is called the 'coefficient of restitution' or COR. This is expressed as a number between 0 and 1⁷, with 0 being no bounce at all, like Goldilocks' beanbag, and 1 being a 'perfect' bounce, up to 100% of the original height. This is only achievable with a hyperelastic polymer, a theoretical, ideally elastic material that 'loses' no energy at all to heat dissipation and will not permanently deform. A regulation lacrosse ball has a COR of 0.6 - 0.7, a regulation basketball has a COR of 0.75, a regulation golf ball has a COR of 0.78, and a regulation ping-pong ball has a COR of 0.94.

In theory, COR values below 0 are possible for a ball that not only doesn't bounce, but sticks, such as a steel ball bearing dropped on a magnetic surface. The only way to get a coefficient of restitution above 1 is to use a ball that frees extra energy when it bounces - by making it out of a contact explosive, for example. Practically, in keeping with the First Law of Thermodynamics, this means that part of the energy will be converted into 'useless' forms on the bounce, so that it takes more energy to bounce a ball into space than to throw it.

Work Smarter, Though Just As Hard

So why are we going to bounce the ball off the floor rather than throw it straight up? In order to have the ball go off into space instead of falling back down to Earth, we need it to achieve escape velocity, which is a whopping 11.25km/s at the Earth's surface. Even the best baseball players have never achieved a pitch faster than 50m/s - less than 1/225th of the required velocity. Escape velocity is lower the farther we get from the Earth's centre of gravity, but using a rocket to take us into the upper stratosphere, where we would need a mere 10.9km/s, is cheating⁸ - and climbing up a tower won't get us high enough to do much. Or will it?

The Potential Of Potential Energy

The beautiful thing about energy is that it can be converted back and forth into various forms - this happens when solar energy is transformed into chemical energy by a tree, which is then compressed into coal by gravitational energy and burned to produce thermal energy, which moves a turbine that converts it into electrical energy, which is piped into your house and turned into sound energy by the radio, so you can listen to the evening news. Human beings don't seem to be built for throwing things at velocities greater than 50m/s, but human beings have legs for climbing stairs and hands for carrying things. Carry a ball to the top of a 128m tall building - about 42 storeys, admittedly a bit of a climb - and let it drop straight down, and it will have accelerated to just over 50m/s when it hits the pavement⁹.

This is not actually less work than simply throwing a ball at 50m/s - physically speaking, it's more work, as you're carrying yourself as well as the ball up the stairs. But it's work that a human is able to do. Unfortunately, it still doesn't help us much, because we'd need a tower that sticks up just a little farther than we want the ball to bounce - and we all know what happened the last time someone tried that.

Getting Kinetic

Of course, we can throw the ball at the ground rather than simply dropping it - dribbling a ball requires accelerating it toward the ground harder than mere gravity would, so that it will bounce back to the same height from which it left or higher. However, even climbing the highest publicly accessible point from which one could drop the ball directly down to a hard surface - the observation platform of the CN Tower in Toronto, Ontario, Canada, which is 446.5m above ground level - and flinging the ball down as hard as anyone possibly can at 49m/s would give us only a final velocity of about 105m/s¹², a far cry from the 11,250m/s that we need.

So if we're already dropping the ball from the highest point we can, and accelerating it as much as we can beforehand, what else can we do to get it to bounce high enough to escape Earth's gravitational pull? Can we give it a little shove from below?

Pushing It

This is where you're going to need both your balls¹³. Bounce each of them on the floor a few times to get an idea of their rebound. Now put the tennis ball on top of the basketball, right over the centre, drop them straight down together from about chest height, and watch the tennis ball sail off way over your head! If you don't believe this, you haven't tried it for yourself... So how does it work?

A Bit Of Relativity, Involving No Bicycling At One-Half Light Speed

And also no trains¹⁴ and the cows that watch them, moving walkways, or other traditional thought experiments for understanding frames of reference - we got into this mess using balls, and we'll use the balls to bounce us back out. Having been dropped from chest height, the basketball will be travelling at about 5m/s when it hits the floor and rebounds, now travelling upward at 5m/s¹⁵ while the tennis ball, if only for fractions of a second, is still going down at 5m/s. Now imagine a colony of tiny aliens - the Basketballinians - on the surface of the larger ball. To them, of course, the basketball is standing still - we don't feel the Earth's motion around the Sun, either. But their instruments indicate that the fuzzy yellow asteroid that is threatening to drop on their heads is approaching at an amazing rate of 10m/s! Luckily, both the tennis ball and the basketball are very elastic, so the Basketballinians watch the impact from a safe distance¹⁶ and see the tennis ball asteroid shoot back up into the sky it came from - at 10m/s. This is where you come in again, as an astronomer, watching the basketball-tennis-ball system from the outside. You see the basketball still moving at 5m/s - and the tennis ball moving 10m/s faster than the basketball, for a grand total of 15m/s! The tennis ball has reversed direction and is now travelling at three times its original velocity, which means that it bounces nine times higher.

In order for this to work properly, two details are important. First, the centres of the balls must be aligned exactly and they must fall straight, because otherwise, the lower ball will only strike a glancing blow and not transfer as much energy - and part of the energy transferred will go to making the top ball spin, or fly off at an angle. The second is that the lower ball must be much more massive¹⁷ than the top ball. That's because the two balls colliding each push the other - but a very light object can't transfer as much of its energy to a much heavier one, and thus gets to keep more. Imagine an ant walking into an elephant - and then an elephant walking into an ant! When you bounce a ball off the ground, that moves the Earth, too - but by so little that the effect is negligible.

Piling Up The Balls

We don't have to stop at stacking two balls. Each ball will bounce up at its own velocity plus twice that of the ball below it - so in our example, the third would be going a respectable 35m/s. The fourth ball will be shooting up at 75m/s, already much faster than anyone could have thrown it. By the fifth ball, at 155m/s, we will already have surpassed the height of the bounce off the CN Tower. The sixth ball will be travelling at 315m/s. The seventh, at 635m/s will break the sound barrier¹⁸ and the eight ball, at 1,275m/s, will be faster than the SR-71 'Blackbird', the fastest airplane in the world. The ninth ball, at 2,555m/s, will be well into the hypersonic¹⁹ range. The tenth will knock into the 11th at 5,115m/s, sending the 11th off at a velocity of 10,235m/s, which in turn will speed the 12th up to 20,475m/s - nearly twice escape velocity, though the balls were only dropped from chest height.

This, of course, assumes a 'perfect' bounce. The balls will be slowed down by such things as air resistance and the fact that they are not perfectly elastic. But it has been calculated²⁰ that a ping-pong ball balanced on nine other balls, each much more massive than the one above it and with a very high coefficient of restitution, could be accelerated to escape velocity when the stack is dropped from a height of five metres - the height of a diving platform.

So What Would Launch Day Look Like?

Imagine a balmy summer evening, around dusk. The sky is clear, and the air is still. The pre-launch party has been going on for a few hours, with good music, beer, and a barbecue, but now the tannoy bleats out that countdown is about to commence, and all eyes turn toward the hill opposite, where an unimposing tower is positioned over a perfectly level bit of tarmac. A few brave geeks in heavy protective gear are making last-minute adjustments to the framework in which the balls are suspended so they will fall exactly straight, a marvellous machine whose design made a pretty good thesis topic for a mechanical engineer. The audience is reminded once more to stand with their fingers in their ears and their mouths open - not only because it makes for a good photo for the assembled press. There is a drum roll, and the spotlight snaps to the podium with the big red button, where someone is waiting for their big moment to press it - depending on the location and on who organised the event, this may be a local celebrity, a member of the royal family, the winner of a charity auction, the mayor of a twinned town, or simply whoever called it first. Cameras are positioned, the figures on the hill scurry to safety behind their protective plate-glass barrier²¹, a light is focussed on the tiny ball at the top of the pile, heavy with signatures, and the countdown begins...

10 - 9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 - Boing!

The balls drop. The balls bounce back up, knocking into each other, all centred one above the other, helped by all the crossed fingers out there. Because we've managed to gain the sponsorship of a government organisation that probably wants to pull one over on the ufologists, the balls are made of an ultra high-grade polymer, and don't shatter into a cloud of shards on impact at the higher velocities like mere rubber would. There is a series of sonic booms²², setting off alarms and barking dogs all over the county - after all, under our launch conditions escape velocity is about Mach 32.5, well into the hypersonic range. Because even our friend with his shiny new engineering doctorate couldn't design something that will drop the balls perfectly straight, the slower ones off the bottom will drop back down and scatter over the hillside in a series of wild bounds, later to be hunted down as souvenirs. The balls in the middle range come down hard enough to leave holes in the tarmac and bury themselves deep in the ground, where they will be found by archaeologists of the future and declared ritual objects. The balls at the top of the pile will be visible as bright, glowing dots in the night sky - friction will have caused them to catch fire. It looks like we shouldn't have ignored air resistance after all! One or two may fall back down into the underbrush and give the attending firefighters something to do besides pulling cars out of ditches. The rest will burn up in the atmosphere, though it's nice to imagine that at least part of the tiny superball on the very top makes it into orbit. Perhaps it has an indestructible ceramic core...

Sounds Cool, But That's Too Much Work²³

If your primary objective is not launching a superball into space, but getting a superball into space, you might try putting it inside a ping-pong ball, so it can join the PongSat programme. JP Aerospace, a volunteer-based space organisation based in California, USA, is developing an 'airship to orbit', technology that will use lighter-than-air vehicles to carry cargo and passengers beyond Earth's atmosphere. During research missions, balloons are flown into the upper atmosphere, and then launch rockets upwards. JP Aerospace allows anyone to join these missions by sending an experiment that will fit inside a ping-pong ball up to the very edge of space - free of charge. The ping-pong balls are then returned to their owners, along with data collected during the flight. Space on the balloons is allocated on a first-come, first-served basis, so design your experiment today!