Newton’s third law: the impulse one ball gives another is exactly equal and opposite, so the total is conserved however the energy is split, and it holds regardless of what you set elasticity to.
Kinetic energy is conserved only at e = 1. Below that, some becomes heat, sound and deformation at every impact; the chart shows how much survives.
With m₁ = m₂ and e = 1 the algebra collapses to an exchange of velocities: not a separate rule, what the general formula reduces to.
Does not mean “stops”: the bodies end up moving together at one velocity. Momentum is shared, not destroyed.
Pull back 2 balls on a 6-ball cradle and release at e = 1: the two end balls stop dead on contact and exactly 2 fly out the far side at the same speed. Not because the cradle is special: equal-mass elastic collisions swap velocities, and touching balls do it in an instant, unbroken chain. Two balls arriving together deliver two velocity handoffs in the same instant, which only have room to land on the last two balls in the chain.
Air resistance and imperfect steel give e just under 1: close enough to look perfect for a few swings, but the small loss compounds visibly over a minute of clicking. Set the slider below 1 and the sharp click turns into a duller, clustered motion where balls move together instead of handing off.
Even at e = 1 a light ball hitting a heavy one mostly bounces back; a heavy ball hitting a light one barely slows. Toggle “vary ball mass” to see the clean click become a mix of different responses.
Walls (in Free Collision) and the pivot bar’s string tension (in Newton’s Cradle) are external forces, exactly as a real wall or support would be. Momentum between the two colliding bodies is exact.
Solving the conservation equations for momentum and kinetic energy simultaneously with m₁ = m₂ has exactly one non-trivial solution: the two velocities exchange. It looks like the balls passed through each other, but they didn't — they just handed off exactly what they carried.
Because the swap happens ball-by-ball down the line, near-instantaneously since they're touching. Two balls arriving together deliver two separate velocity handoffs in the same instant, which only has room to land on the last two balls in the chain — that's the whole trick, not a separate rule.
Momentum still balances exactly, but each impact bleeds off some kinetic energy — set the slider below 1 and watch the sharp cradle "click" turn into a duller, more clustered motion where balls tend to move together rather than cleanly handing off velocity.
Yes — unequal masses never fully swap velocities even at e = 1. A light ball hitting a heavy one mostly bounces back; a heavy ball hitting a light one barely slows. Toggle "vary ball mass" to see the clean cradle click become a mix of different responses.
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