M048 · Stored energy takes flight

Spring Launcher

Compress a spring, release a ball and follow the energy into flight. Find the launch threshold, compare gravity and recover stiffness from recorded apex heights.

Contact state $\text{Ready}$

Ball-bottom height $\text{Ready}$

Upward velocity $\text{Ready}$

Predicted highest point $\text{Ready}$

Compression needed for flight $\text{Ready}$

Stiffness from recorded heights $\text{Ready}$

Energy ledger residual $\text{Ready}$

Recorded apex heights $\text{Ready}$

SET / RELEASE / MEASURE

Make a prediction. Then let go.

Initial compression

c

0.32\,m

Spring stiffness

k

120.00\,\frac{\mathrm N}{\mathrm m}

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\,

The ball rests on a massless compression spring and is not attached to it. The spring pushes only while compressed. Its relaxed tip is 0.8 metres above the floor. Gravity acts throughout the spring stroke. Below the flight threshold the ball stays in contact and oscillates; the model has no friction, spring mass, damping or mechanical stop.

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.

Stiffness inference uses only recorded peaks above the relaxed spring tip. If no such height is resolved it reports unavailable. A near-threshold noisy peak can bias this classification. Ideal spring and gravitational energy assumptions are required.

Exact unilateral spring and ballistic motion; no air resistance or spring mass.

Start with the essentials

Focus question: Will the ball actually leave the spring?
One-sentence intuition: Gravity takes energy during the spring stroke before flight even begins.

Core mathematical model

Contact and flight

m\ddot q=-k\min(q,0)-mg

The coordinate starts at the relaxed spring tip; the spring never pulls.

Flight threshold

c>\frac{2mg}{k}

A strictly positive launch speed requires more than static spring support.

Apex above the floor

H_{\max}=L-c+\frac{kc^2}{2mg}

Valid only when separation occurs. Height is ball-bottom clearance.

Recorded-height inference

\widehat k=\frac{2mg}{c^2}(\overline H_{\rm peak}-L+c)

Only recorded peaks above the relaxed tip enter this ideal inference.

Common difficulties

Read the boundary

Typical misconceptionAny compression guarantees flight.

Better mental modelCompare the weak-spring preset with the launch threshold.

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

Contact and flight

Flight threshold

Apex above the floor

Recorded-height inference

Common difficulties

Read the boundary

Run the experiment

Set the height

Follow the event

Keep an energy ledger

Measure a run