elastic inelastic collision explained
When two bodies collide, the total momentum before and after always stays the same. How much they bounce apart, though, differs from body to body: a rubber ball springs back hard, while a lump of mud splats and sticks. The single number that captures this difference is the coefficient of restitution e, which ranges from e=1 for a perfectly elastic collision to e=0 for bodies that stick together. Momentum is always conserved, but depending on e, part of the kinetic energy escapes as heat and sound. In this lesson you learn to connect before and after with two equations, one for momentum and one for restitution.
Ball 1 approaches ball 2, which is at rest. The upper row is before the collision and the lower row after; the relative speed at which the balls close in is the approach speed, and the relative speed at which they separate afterward is the separation speed. Change the approach speed v with the slider: in this case, perfectly elastic with equal masses, the blue ball stops after the collision and the orange ball leaves at the same speed. The two balls have simply swapped velocities, and the readout confirms that the separation speed equals the approach speed. It is exactly what happens when one billiard ball hits another head-on: the cue ball stops and the struck ball rolls on.
Even with unequal masses, the total momentum before and after a collision is the same. Change the mass m₁ of the blue ball with the slider and the post-collision velocities in the readout shift around, but the two green bars below, the total momentum before and after, always have the same length. A heavy ball striking a light one sends both forward; a light ball striking a heavy one may bounce back, yet in every case the sum is unchanged. No force enters from outside, and the forces the two balls exert on each other are equal and opposite, so whatever momentum one gains, the other loses exactly.
The coefficient of restitution e, which measures how bouncy a collision is, is defined as e = separation speed ÷ approach speed. The upper blue arrow is the approach speed, held fixed, and the lower orange arrow is the separation speed. Drag e with the slider and the orange arrow adjusts to e times the length of the blue one. At e=1 the two arrows match and the bodies bounce off as fast as they came; at e=0 the orange arrow disappears and the bodies move on stuck together. Real objects fall somewhere in between: a lively ball is close to 1 and modeling clay close to 0.
Switch between collision types with the buttons and the differences stand out. A perfectly elastic collision (e=1) conserves even kinetic energy, so the bar is all green; a partially elastic one (0<e<1) loses some and a red segment appears; a perfectly inelastic one (e=0) loses the most as the bodies stick together. Momentum, however, is conserved in every case. Look at the blue ball's arrow after the collision: in the elastic case it stops, and when the balls stick it moves on with the orange ball at half speed. How much kinetic energy survives is what tells the types apart.
Drag e and you see how the remaining kinetic energy shrinks. For a head-on collision between equal masses with one initially at rest, the fraction kept is (1+e²)2, so at e=1 everything is conserved and at e=0 only half remains. The green segment is the share kept and the red segment the share lost, which scatters as heat, sound, and deformation. Lowering e a little costs little at first, but the smaller the coefficient becomes, the faster the red segment widens. Cars are built with crumple zones for exactly this reason: to absorb kinetic energy in the deformation of the body.