Marble Collision Lab (2D): Impact Parameter, Restitution and Spin Throw

What is simulated

The same rigid-sphere engine as the 3D page (solid marbles with position, velocity and 3D spin, a fixed 0.8 ms step), seen from above. The top view shows what 3D hides: the felt-slip arrow (red), the velocity arrow (blue), marks that turn with the marble, and a lab graph of how the angle between two marbles depends on where they meet.

Collision lab

For a glancing hit of equal marbles whose centres are offset by an impact parameter b (in units of 2R), the line of centres makes the angle phi = asin b with the shooter's path. Only the velocity component along it, v_n = v cos phi, is exchanged: v2' = (1+e)/2 v_n n and v1' = v - v2'. With e = 1 the two marbles leave exactly 90 degrees apart for every b; with e < 1 the angle opens up. The graph compares this closed form (line) with the engine impulse code (yellow dots). Pink dots add Coulomb friction between the marbles, which drags the target sideways (the "throw") and bends the shooter.

Friction on the felt

The slip velocity is u = v - R (w x z). While u is not zero the felt applies F = -mu m g u/|u|; for a solid sphere du/dt = -(7/2) mu g u_hat, so sliding ends after 2|u0|/(7 mu g) and a marble kicked at its centre rolls at 5/7 of its first speed. Rolling is then slowed only by rolling resistance, a constant deceleration mu_r g.

The kick

An impulse J at height h and left offset a gives v = J/m and w = (5J/(2 m R^2)) (h e_left - a z). At h = 2R/5 the marble rolls at once (v = wR); a lower tip gives backspin (draw), and the shooter can stop dead on contact.

Impacts and cushions

Normal impulse Jn = (1+e) m_red v_n; tangential Coulomb impulse Jt = min(mu_b Jn, m_eff |u_t|) with 1/m_eff = (1/m1 + 1/m2)(1 + 1/0.4). Cushions use the same rules against an immovable wall.

Simplifications

• Marbles stay on the table: no bounce, no vertical contact friction, no deformation of the felt.
• Friction coefficients and restitution are constants; real values depend on speed and surface.
• Impacts are instantaneous and resolved one pair at a time.
• Spin marks are a drawing aid: four points on the sphere that rotate with the integrated 3D spin.

Checks

Six boxes under the controls run the page's own functions against closed forms: slide-to-roll, head-on velocity exchange, the 90 degree rule, vertical angular momentum with friction, the centre of percussion, and the angle law above across 54 combinations of b and e.