Domino chain - 2D side view and wave speed

One domino

A domino of height h, thickness t and mass m tips about its front bottom edge. Its moment of inertia about that edge is I = m(h² + t²)/3 and gravity gives the torque m g (h sinθ − t cosθ)/2, so the equation of motion is I θ'' = m g (h sinθ − t cosθ)/2. A standing domino is stable until tanθ > t/h, and from there it accelerates, converting potential energy into rotation: ½ I ω² = U(θ₀) − U(θ) with U = m g (h cosθ + t sinθ)/2.

The chain

When the top corner of domino i reaches the back face of domino i + 1 (after tipping by about sin−1(s/h) for gap s) the two collide. Here the contact is a stiff damped spring acting at the top edge, so a damping ratio sets how much energy each collision loses. The tipping of each domino adds energy to the pulse, so a steady toppling front of speed v ≈ (s + t)/Δt forms. Dimensional analysis gives v ∝ √(g h) for geometrically similar chains, so a chain scaled up four times travels twice as fast; small gaps give many weak pushes and large gaps give fewer, harder ones, with a maximum speed in between.

Honest simplifications: rigid thin plates, a single contact point at the top edge, no floor friction or sliding, no air, dominoes that cannot tip backwards, and a penalty-spring contact whose stiffness is scaled with m g / h. Real chains also show bouncing and slip that this model leaves out, so absolute speeds are indicative while the scaling is the honest result.

• Press Wide gaps: the pulse slows, and beyond a gap of about 0.9 h it stalls.
• Press Tight gaps: dominoes lean on each other and the wave is slow but dependable.
• Use the Giant dominoes preset and compare the wave speed with the standard chain: it is close to twice as fast for 4 times the size.
• Raise the collision damping and see the wave speed fall.