M050 · Every bounce tells a story

Bouncing Ball & Restitution

Where does the missing height go? It is lost in each impact. The coefficient of restitution is the rebound speed divided by the impact speed, and the height a ball reaches grows with its speed squared. Each new peak is the previous one times the coefficient squared, so a ball with a coefficient of 0.9 keeps 81 percent of its height on every bounce.

Drop a ball, watch the impact in slow motion and compare elastic, lossy and sticking collisions. Record apex heights to estimate the coefficient of restitution, and see how the ideal model fits infinitely many bounces into a finite time.

Motion state $\text{Ready}$

Ball-bottom height $\text{Ready}$

Upward velocity $\text{Ready}$

Energy transferred at impacts $\text{Ready}$

Ideal collapse time $\text{Ready}$

Restitution from recorded heights $\text{Ready}$

Energy balance residual $\text{Ready}$

Recorded apex heights $\text{Ready}$

Release the ball and measure its motion

Make a prediction, then let go

Release height above floor

H_0

2.50\,m

Normal restitution

e

0.75\,

Ball mass

m

0.50\,kg

Gravitational acceleration

g

9.81\,\frac{\mathrm m}{\mathrm s^2}

Replay window

T

6.00\,s

Bounded apex-height error

\epsilon

0.00\,m

Sensor error seed

s

17.00\,

Height is the gap between the bottom of the ball and a fixed horizontal floor. Gravity is uniform. Each impact takes no time and reverses the normal velocity, scaled by a constant coefficient of restitution. The energy the ball loses goes into the floor and into internal modes the model does not resolve. Finite peak force, deformation, spin, drag and floor motion are not modeled.

Gold is the ball, and blue marks the relaxed spring tip or the floor. All lengths in the main scene share one true scale, and the ball radius is 0.14 metres. The faint spheres are earlier positions from the model, not measurements. Follow view moves with the ball, and Overview shows the whole height range. Nothing plays automatically.

Recording uses an ideal trigger at each top point, with exact timestamps and a bounded, repeatable height error. It covers the chosen replay window, whatever the current replay time. The CSV holds only the peak number, time and noisy height. Replaying keeps the record; changing a parameter clears it. The fits read only the recorded heights and the given constants. The error is not a calibrated uncertainty model.

When the restitution is strictly between zero and one, the flight times form a geometric series, so infinitely many impacts happen in a finite time. The ideal model then continues with the ball at rest; this does not predict what real contacts do at the microscopic scale. Only the first 100 resolved impacts are listed. The height fit drops pairs smaller than five error bounds, and it is biased when both coordinates are noisy.

Impacts are instantaneous and the floor does not move. The ideal model's infinite run of bounces in a finite time marks the edge of the model.

Start with the essentials

Focus question: Where does the missing height go?
One-sentence intuition: The ratio of successive bounce heights is the square of the speed ratio.

Core mathematical model

Instantaneous impact

v^+=-e v^-,\qquad J=m(v^+-v^-)

The floor takes the opposite impulse; the ball alone does not conserve momentum.

Apex height and energy after each bounce

H_n=e^{2n}H_0,\qquad E_n=mgH_n

Uniform gravity and no drag are assumed between instantaneous impacts.

Infinitely many bounces in a finite time

t_*=\sqrt{\frac{2H_0}{g}}+\frac{2e\sqrt{2gH_0}}{g(1-e)}

With zero restitution the first impact stops the ball. With a restitution of one, the bounces never die out.

Restitution from recorded heights

\widehat e=\sqrt{\frac{\sum_n H_nH_{n+1}}{\sum_n H_n^2}}

The fit is a line through the origin, using resolved pairs of successive recorded heights. Noise in both heights of a pair can bias the estimate.

Common difficulties

The floor pushes from outside

Typical misconceptionThe ball conserves momentum when it hits a fixed floor.

Better mental modelThe floor gives the ball an external impulse, so the ball's momentum changes. Count the energy transferred at impact as a separate term.

Run the experiment

01
Set the height
Drag the gold handle or use its keyboard slider.
What to observe: The scene and predicted motion update together.
02
Follow the event
Play, pause and jump to the next event.
What to observe: Height changes smoothly, but an impact can change the velocity instantly.
03
Track the energy
Drag the time between turning points.
What to observe: Stored, kinetic, gravitational and transferred energy always add up to the same total.
04
Measure a run
Record apex heights, export the CSV and compare the inferred parameter.
What to observe: Replaying preserves the same measurements; changing physical parameters clears them.

Core mathematical model

Instantaneous impact

Apex height and energy after each bounce

Infinitely many bounces in a finite time

Restitution from recorded heights

Common difficulties

The floor pushes from outside

Run the experiment

Set the height

Follow the event

Track the energy

Measure a run