M048 · Stored energy takes flight

Spring Launcher

Will the ball actually leave the spring? Only if the spring is squeezed far enough. While the spring pushes, the ball is already rising, so gravity takes part of the stored energy before flight begins. The ball flies off only if the compression is more than twice the sag the ball's weight alone would cause; below that, it just rides up and down with the spring.

Compress a spring, release a ball and follow its energy into flight. Find the smallest compression that launches the ball, compare different gravity and work out the spring 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 balance residual $\text{Ready}$

Recorded apex heights $\text{Ready}$

Release the ball and measure its motion

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 sits on a massless compression spring without being attached, so the spring pushes only while compressed. The relaxed spring tip is 0.8 metres above the floor. Gravity acts over the whole spring stroke. Below the flight threshold the ball stays on the spring and oscillates. The model has no friction, spring mass, damping or mechanical stop.

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.

The stiffness estimate uses only recorded peaks above the relaxed spring tip. If no such peak is found, the estimate is reported as unavailable. A noisy peak close to that height can be sorted onto the wrong side. The estimate assumes an ideal spring and ideal gravitational energy.

The spring can only push, and both the spring stroke and the flight are solved exactly. Air resistance and the mass of the spring are ignored.

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 is measured from the tip of the relaxed spring, and the spring never pulls.

Flight threshold

c>\frac{2mg}{k}

The ball leaves with a positive speed only if the compression exceeds twice the sag of the spring holding the ball at rest.

Apex above the floor

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

This holds only if the ball leaves the spring. Height is measured from the floor to the bottom of the ball.

Stiffness from recorded heights

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

This ideal estimate uses only recorded peaks above the relaxed spring tip.

Common difficulties

Not every compression launches

Typical misconceptionAny compression guarantees flight.

Better mental modelCompare the weak-spring preset with the launch threshold. Below the threshold the ball never leaves the spring.

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

Contact and flight

Flight threshold

Apex above the floor

Stiffness from recorded heights

Common difficulties

Not every compression launches

Run the experiment

Set the height

Follow the event

Track the energy

Measure a run