Wrecking-Ball Impact Lab (2D)

Pendulum swing

The ball hangs on a rigid rope of length L and moves on a circle: d²θ/dt² = -(g/L) sinθ. This page integrates it with a fourth-order Runge-Kutta step of 1 ms (not the small-angle shortcut). Energy conservation gives the impact speed v = sqrt(2 g L (1 - cos θ0)); the first check compares the integrated value with this formula.

Impact with restitution

For a ball (M, v) hitting a free crate (m, at rest) head-on: vcrate = (1+e) M v/(M+m), vball = (M - e m) v/(M+m). Momentum is conserved exactly; the kinetic energy lost is ½ μr (1 - e²) v² with reduced mass μr = M m/(M+m). e = 1 is perfectly elastic, e = 0 perfectly plastic.

Sliding

Coulomb friction decelerates the crate at μ g, so it stops after d = vcrate² / (2 μ g). The friction work μ m g d equals the crate's kinetic energy, which closes the ledger.

Simplifications

One free crate, no rope mass, no air drag, an instantaneous impact described by one restitution coefficient, steel ball density 7850 kg/m³. The 3D lab uses a wall of crates with breakable joints; this 2D view isolates the momentum and energy bookkeeping. Educational pendulum-collision physics only.

What to try

• Drag the ball to a new release angle and watch the impact speed follow the curve in the second graph.
• The heavy-and-slow and light-and-fast presets carry equal energy, but the heavy ball transfers more momentum.
• Compare the energy lost in the impact for e = 0 and e = 0.9.