Cushion vs Wall · SimulatorSame Impulse, Different Peak Forces
Same ball at same speed hits a wall, then a cushion; same impulse, different peak forces and times.
Published: August 23, 2026
Objective
Verify the impulse-momentum theorem by comparing two elastic collisions that share the same impulse but differ in contact time. With a fixed ball mass and speed, the total momentum change is constant; this sim shows that spreading the same impulse over a longer contact time (cushion) reduces peak force in direct proportion, leaving the shaded area under the force-time curve identical.
Setup
- Leave sliders at their defaults: mass 0.5 kg, impact speed 8 m/s, wall contact time 0.01 s, cushion contact time 0.2 s. Read the Impulse readout — it should show 8.00 N·s.
- Press Start. Both balls travel right simultaneously, hit their surfaces, and rebound. Watch the force-time graph on the right panel: a narrow crimson spike (wall) and a wide green pulse (cushion) draw out in real time.
- Once both balls return, note the Wall Peak F and Cushion Peak F readouts. Record the Force Ratio (~20.00). Compare the two shaded areas under the curves — they are visually equal.
- Press Reset. Change the wall contact time to 0.005 s (its minimum). Press Start again. The wall spike becomes even taller while the impulse readout stays at 8.00 N·s.
- Reset and increase the cushion contact time to 0.5 s (its maximum). Press Start. The cushion peak shrinks further; record the new force ratio.
Analytical Prediction
For a fully elastic rebound, impulse J = 2·m·v. With m = 0.5 kg and v = 8 m/s:
Peak force follows the half-sine pulse model F_peak = (π/2)·J/Δt:
The ratio equals the inverse of the contact-time ratio exactly, because both peak forces share the same J in the numerator. The Impulse readout must remain 8.00 N·s regardless of how the contact-time sliders are moved.
Results Analysis
After a completed run with default sliders, the simulation readouts should show: Impulse (N·s) = 8.00, Wall Peak F (N) ≈ 1256.6, Cushion Peak F (N) ≈ 62.8, Force Ratio ≈ 20.00. These match the analytical prediction to within the 1-decimal rounding shown in the HUD. The force-time graph confirms the equal-area result visually: both shaded regions represent 8.00 N·s, yet the crimson wall spike reaches about 20 times the height of the green cushion pulse. The dashed reference lines at each peak force value make this height difference easy to read. Changing the wall contact time slider to 0.005 s doubles the wall peak to ~2513 N while the cushion peak is unchanged and the impulse stays at 8.00 N·s.
Source of Error
The simulation assumes a perfectly elastic rebound (coefficient of restitution = 1), so kinetic energy is fully conserved and the ball exits at the same speed it entered. Real impacts are partially inelastic: some energy is lost to heat, sound, and permanent deformation, so the true impulse is slightly less than 2·m·v. The half-sine pulse shape is a standard engineering model for smooth impacts; real force profiles are asymmetric and noisier. Air resistance during the ball's approach and rebound is neglected. The cushion is modeled as a linearly compliant surface with a fixed contact time independent of force magnitude, whereas real foams and air bags have a force-dependent spring characteristic. These physical idealizations are the same ones the analytical prediction assumes, so the residual gap between prediction and readouts is numerical rather than physical.
Further Exploration
- Set wall contact time to its minimum (0.005 s) and cushion contact time to its maximum (0.5 s). What force ratio does the readout show? How does that compare to the ratio of the contact times?
- Double the ball mass (from 0.5 kg to 1.0 kg) and press Start. Does the force ratio change? Why or why not? What happens to the impulse readout?
- Set the impact speed to 1 m/s (its minimum). How does the graph y-axis scale? Is the force ratio still 20.00? What does this tell you about the universality of the cushion principle?
- After one completed run, press Reset (keeping the board) and change only the wall contact time before running again. Can you overlay curves to show three wall peaks at three different contact times, all with the same shaded area?
- Why do car air bags inflate so rapidly (in about 30 ms) rather than remaining permanently inflated? Use the cushion contact time slider to explore how the benefit decreases once the contact time grows beyond the cushion's useful range.