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The science
Newton's third law, said carefully
Forces come in pairs: if body A pushes on body B with force FBA (the force on B by A), then B pushes on A with FAB = −FBA, at the same instant, along the same line, of the same kind (spring with spring, magnet with magnet, contact with contact). The two forces act on different bodies, which is why they never cancel inside one body's equation of motion: cart A accelerates under the force it feels, cart B under the opposite one. What does cancel is their effect on the pair as a whole.
Consequences you can watch here
Momentum. With only internal forces d(pA + pB)/dt = FAB + FBA = 0, so the total momentum and the centre-of-mass velocity vcm = (mAvA + mBvB)/(mA + mB) stay constant. The accelerations are inversely proportional to the masses, aA/aB = −mB/mA: the light cart always gets the big acceleration. Push-off. A spring released between two carts at rest gives vA = −√(2E mB / (mA(mA+mB))) and vB = +√(2E mA / (mB(mA+mB))) with E = ½ k x2 the stored energy; the lighter cart takes most of the energy. Collisions. For an elastic head-on collision vA' = ((mA−mB)vA + 2mBvB)/(mA+mB); with restitution e the relative speed after is e times the relative speed before and kinetic energy is lost, but the momentum is not. Impulse. The areas under the two force curves are equal and opposite and equal the momentum changes (∫F dt = Δp).
What is not a pair
The end walls and the track friction are external forces: the wall pushes on a cart and the cart pushes on the wall (the pair partner, which is the whole laboratory and is not tracked), so the carts' total momentum is no longer conserved when the walls are on or mu > 0. The orange arrows show these forces. Friction is Coulomb friction regularised as −μ m g tanh(v/0.02).
Model and honest simplifications
One-dimensional carts of half-length 0.15 m. Spring plunger: F = k (l0 − gap) while the gap is shorter than its free length l0 (the plunger only pushes); tied spring: F = −k (gap − l0); magnets: repulsion K/gap2 (point-pole approximation, no torques). Bumpers and walls are stiff spring-dampers (4000 N/m) whose damping is set from the restitution through ζ = −ln e / √(π2 + ln2 e), with the force clamped so a bumper only pushes; contact lasts a few milliseconds instead of zero. The equations are integrated with the symplectic velocity-Verlet scheme at 0.5 ms. The pair force is computed once and applied to A and B with opposite signs, so the page never checks the law against itself: the four in-page checks test momentum drift in a lossy collision, the push-off speeds, the elastic-collision formulas and the impulse-momentum theorem, none of which is built into the force routine. Air-track drag, wheel inertia and sensor noise are not modelled.