Why Does a Ball Bounce Lower Each Time?
– It Isn't Lost, It Slowly Turns into Heat and Sound
A dropped ball bounces lower the second time, and lower still the third. It looks as if some force has "run out" somewhere, but nothing has actually vanished. Each time the ball squashes against the floor and springs back, it turns some of its motion into heat and sound and lets it go. And it gives up the same share every time.
A child drops a bouncy ball on the floor. Boing, boing, boing, bob, bob-bob-bob... It bounces lower each time, the gaps between the sounds get shorter and shorter, and at last it comes to rest, quivering on the floor.
"Hey, why does it get lower and lower? Why can't it just keep bouncing the same height?" Asked that, you might want to answer, "Because it runs out of power."
But where did that "power" go? Touch the ball and nothing seems to be missing. The answer is hidden in the split second of each bounce.
There Are Only Two Reasons It Gets Lower
When the ball hits the floor, it flattens for an instant and then springs back up as it returns to shape. But the push it gives back on the way out is a little smaller than the push it took in. The difference turns into heat inside the ball.
The "pop" you hear is proof that the ball did work shaking the air. The floor also trembles slightly. All of this is energy shared out from the ball's motion.
In both cases, the ball's motion has simply changed into another form, and the total hasn't shrunk. Science calls this "conservation of energy." But the part that became heat and sound can no longer help lift the ball.
It Drops by the Same Share Every Time
A bouncing ball follows a curious rule. It isn't the number of centimetres lost that stays the same, but the fraction left, which is about the same every time. Drop a table tennis ball on a hard table and it rebounds to roughly three quarters of the drop height. From one metre, that's 75 cm, then 56 cm, then 42 cm. The height shrinks by multiplication.
As the height falls, the time in the air gets shorter too. That is why the sound goes "boing, boing, bob-bob-bob," with the gaps closing up. In Figure 1, try changing the rebound share with the slider.
Here is the odd part. On paper the ball bounces infinitely many times, yet the time until it stops is finite. The humps shrink so fast that adding them all up takes only a few seconds. In reality, at the very end you can't tell whether the ball is still bouncing or just trembling.
The Lost Share Becomes Heat Inside the Ball
When a rubber ball squashes, the long rubber molecules inside rub against each other as they stretch and shrink. Just as rubbing your hands together warms them, this rubbing makes heat. Look at Figure 2. During the squash and the rebound, the rubbing eats away a little of the ball's push.
The amount that becomes heat is tiny, and you can't feel any warmth by touching the ball. Still, it adds up over repeated bounces. The less a material loses this way, the higher it bounces. A bouncy ball bounces so well because it is made of rubber that rubs very little inside even when squashed.
A cold squash ball hardly bounces at all. Before a match, players hit it against the wall many times to warm it up. The stiffness and internal rubbing of rubber change with temperature, and a warm ball rebounds better. In effect, they put the heat made by each bounce to good use.
Official table tennis and tennis balls have a standard for how high they must rebound when dropped from a set height. If the bounce varied from ball to ball, a match would be unplayable. In the fold-out below, we calculate using the numbers from these standards.
Summary
A ball bounces lower each time because, in the instant it squashes and springs back, part of its motion turns into heat and sound. Since the share lost is nearly the same every time, the height shrinks by multiplication and the gaps between sounds close up. The energy hasn't vanished. It has only changed form and spread into the floor and the air around it.
With every bounce, the ball pays out a little heat.
The falling height is the record of those payments.
Motion turning into heat also comes up in Why do your hands get warm when you rub them together? and Is the water at the bottom of a waterfall warmer than before it fell? For a story about stones bouncing over water, see Why does a stone skip many times across water?
- Stick a tape measure on a wall and gently drop a ball from a height of one metre. Filming from the side with a smartphone makes it easier to read the rebound height.
- Read the heights of the first and second bounces, then compare "first bounce ÷ 1 metre" with "second bounce ÷ first bounce." The two shares should come out nearly the same.
- Compare different balls, such as a bouncy ball, a tennis ball and a ping-pong ball. Try a ball chilled in the fridge against one warmed in your hands.
On a soft floor such as a tatami mat or a carpet, the floor soaks up the energy as heat, so no ball bounces well. Compare on a hard floor. The trick is to let go of the ball gently rather than throw it down.
For those who want to know more – terms, formulas and links to textbooksEach part is labelled by level, from middle school science to university courses
- Middle schoolCovered in middle school science
- High schoolCovered in high school "Basic Physics" and "Physics"
- High school+Advanced high school material, or textbook sidebars
- UniversityUniversity courses not taught in high school (mechanics, strength of materials, polymer physics)
- ResearchNot yet taught as settled fact, even at university; researchers are still investigating it
Middle schoolTerms: this phenomenon has names
- Mechanical energy: the sum of height energy (potential energy) and speed energy (kinetic energy). While the ball is in the air, it stays almost constant.
- Conservation of energy: the rule that energy may change form, but the total neither grows nor shrinks. The ball seems to lose energy at each bounce because it changes into heat and sound.
- Coefficient of restitution (bounce coefficient): the ratio of the speed just after impact to the speed just before. At 1 the ball returns all its speed, and at 0 it does not bounce at all.
Middle schoolHigh schoolChecking with formulas: how much height is lost per bounce
Using the standard values for official table tennis and tennis balls, we calculate the rebound height and the time to stop.
| In symbols | hn ＝ h0 × e2n (coefficient of restitution e ＝ √( h1 ÷ h0 )) |
| In words | Height of the n-th bounce ＝ drop height × (coefficient of restitution squared) multiplied n times |
| Where it comes from | While in the air, mechanical energy is conserved, so height is proportional to the square of speed. Each time the speed at the floor becomes e times as large, the height becomes e squared times as large. |
| Symbol | Meaning and unit |
|---|---|
| h0 | Drop height (cm) |
| hn | Height of the n-th bounce (cm) |
| e | Coefficient of restitution (no unit, between 0 and 1) |
| t0 | Time for the first fall to the floor (s) |
| Table tennis ball: drop height (international rule standard) | 30.5 cm |
| Table tennis ball: approximate rebound height | about 23 cm |
| Tennis ball: drop height (international rule standard) | 254 cm |
| Tennis ball: middle of the rebound height standard | about 141 cm (135–147 cm) |
| Time to fall from a height of 1 m | about 0.45 s |
| Table tennis ball: share of height regained (e squared) | 23 ÷ 30.5 ≒ 0.75 |
| Height of the 1st bounce from 1 m | 100 × 0.75 ＝ 75 cm |
| Height of the 2nd bounce | 75 × 0.75 ≒ 56 cm |
| Height of the 3rd bounce | 56 × 0.75 ＝ 42 cm |
| Tennis ball: share of height regained | 141 ÷ 254 ≒ 0.56 |
| Share a tennis ball loses in one bounce | 1 − 0.56 ＝ 0.44 |
So a table tennis ball gives up about a quarter of its mechanical energy as heat and sound with each bounce, and a tennis ball gives up more than 40%.
| Formula for the time to stop (the result of adding up the hump widths as a geometric series) | T ＝ t0 × (1 ＋ e) ÷ (1 − e) |
| Square root of 0.75 (coefficient of restitution e) | taken as about 0.866 |
| Numerator | 1 ＋ 0.866 ＝ 1.866 |
| Denominator | 1 − 0.866 ＝ 0.134 |
| Divide numerator by denominator | 1.866 ÷ 0.134 ≒ 13.9 |
| Time to stop | 0.45 × 13.9 ≒ 6.3 s |
On paper the bounces go on forever, but the total time levels off at about 6 seconds. This is the same value as the number at upper right in Figure 1. In reality, air resistance and changes in how the ball bounces in slow impacts matter, so treat this as a rough guide.
High schoolHigh school+The coefficient of restitution and the energy lost
High schoolIn Basic Physics, the coefficient of restitution with the floor is taught as "speed after impact ÷ speed before impact." Kinetic energy is proportional to the square of speed, so the share of energy left after one impact is e squared. This matches the share of the rebound height.
High school+The time to stop is a geometric sequence in which the hump width shrinks by a constant ratio e. That is why adding infinitely many terms still gives a finite value. It is exactly the formula for the sum of an infinite geometric series in mathematics.
UniversityViscoelasticity and hysteresis loss
Rubber is a viscoelastic material, with both spring-like elasticity and honey-like viscosity. The force curve while compressing does not match the curve while releasing, and the area between them is the hysteresis loss, which becomes heat. The deformation at the moment of contact is estimated with Hertzian contact theory. The coefficient of restitution is known to change with impact speed and temperature.
📖 Derivations and further reading: Coefficient of restitution (Wikipedia) / Conservation of mechanical energy (Wikipedia)
ResearchWhat is still not well understood
- How the coefficient of restitution changes with speed. It is not constant, and tends to get smaller the faster the impact. Predicting the detailed changes for each material from molecular motion is still a hard problem.
- The nature of the final "trembling." When the bounce height becomes very small, vibrations of the ball and floor and the effect of the thin air layer can no longer be ignored. Where the ball stops bouncing is not decided by a simple formula.
- When many particles collide. In a crowd of grains such as sand, collisions that each lose a little speed repeat, and the grains can suddenly clump together. This "inelastic collapse" is a research topic in granular physics.
In other words, this article too is "an explanation of what is known so far." The rule that the height shrinks by the same share each time should be taken only as a good approximation.
Links to textbooks (by level)
| Level | Subject and unit | Where in this article |
|---|---|---|
| Middle school | Science: "Mechanical energy," "Changes of energy" | The explanation that motion changes into heat and sound |
| High school | Basic Physics: "Coefficient of restitution," "Mechanical energy" | The calculation: share of height ＝ e squared |
| High school+ | Mathematics: "Infinite geometric series" | Why the time to stop is finite |
| University | Strength of materials, polymer physics: "Viscoelasticity," "Hysteresis" | How heat is made inside rubber |
| Research | Contact mechanics, granular physics | Speed dependence of the coefficient of restitution, inelastic collapse |
| – | Connection to daily life | Ball testing in sports, the habit of warming up a squash ball |
- International Table Tennis Federation (ITTF), Laws of Table Tennis and technical documents (ball specification: rebound when dropped from 30.5 cm)
- International Tennis Federation (ITF), appendix to the Rules of Tennis (ball specification: rebound of 135–147 cm when dropped from 254 cm)
- Cross, R. "The bounce of a ball", American Journal of Physics 67, 222 (1999)
- Japanese high school "Basic Physics" (物理基礎) textbooks (units on the coefficient of restitution and conservation of mechanical energy)
*This article is a popular-science explanation for general readers. The figures given are rough guides to help you understand the mechanism. How a ball bounces varies greatly with its material, temperature and the hardness of the floor.