M050 · Every bounce tells a story

Bouncing Ball & Restitution

Drop a ball, slow down the impact and compare elastic, lossy and sticking collisions. Record apex heights to estimate restitution and explore the finite-time ideal collapse.

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 ledger residual $\text{Ready}$

Recorded apex heights $\text{Ready}$

SET / RELEASE / MEASURE

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 clearance of the ball bottom above a fixed horizontal floor. Gravity is uniform. Impacts have zero duration and reverse the normal velocity with a constant restitution. Lost ball energy is transferred to the floor and unresolved internal modes; no finite peak force, deformation, spin, drag or floor motion is modeled.

Gold is the ball; blue marks the relaxed spring tip or floor. Main-scene lengths share one metric scale and the ball radius is 0.14 metres. Faint spheres are earlier model positions, not measurements. Follow view moves with the ball; Overview shows the entire height range. No automatic playback.

Record uses an ideal apex trigger with exact timestamps and bounded deterministic height error. It records the selected replay window, independently of the current replay clock. The CSV contains only peak index, time and noisy height. Replay preserves the record; parameter changes clear it. Fits read recorded heights and supplied constants only. Error is not a calibrated uncertainty model.

For restitution strictly between zero and one, flight intervals form a geometric series. Infinitely many impacts accumulate in finite time. The ideal continuation is rest; this is not a prediction of microscopic real contacts. Only the first 100 resolved impacts are listed. Height inference rejects pairs smaller than five error bounds and is biased when both coordinates are noisy.

Instantaneous impacts on a fixed floor; ideal collapse is a model boundary.

Start with the essentials

Focus question: Where does the missing height go?
One-sentence intuition: A speed ratio becomes a squared height 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 and energy sequence

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

Uniform gravity and no drag are assumed between instantaneous impacts.

Finite accumulation time

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

For zero restitution the first impact ends motion; unit restitution never collapses.

Recorded-height inference

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

A through-origin height-ratio fit uses resolved recorded pairs. Noise in both heights can bias it.

Common difficulties

Read the boundary

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

Better mental modelThe floor supplies an external impulse; account for transferred energy separately.

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 remains continuous while an impact can change velocity instantly.
03
Keep an energy ledger
Scrub between turning points.
What to observe: Stored, kinetic, gravitational and transferred energy explain each other.
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 and energy sequence

Finite accumulation time

Recorded-height inference

Common difficulties

Read the boundary

Run the experiment

Set the height

Follow the event

Keep an energy ledger

Measure a run