Elastic and inelastic collisions

In brief

Roll one billiard ball into an identical, stationary one, dead center, and something striking happens: the moving ball stops completely, and the ball it struck rolls away at the exact speed the first one had. Now press two lumps of soft clay together in the same way and they do not bounce apart at all, they stick and move off together, slower than the original speed. Both events obey the same conservation law, the combined momentum of the two objects is exactly the same before and after the collision in both cases, because that only requires the equal-and-opposite contact forces from the previous entries. What differs is a second quantity, kinetic energy, which survives the billiard-ball collision intact but does not survive the clay collision at all. Telling these two accounts apart, and knowing which one a given collision follows, is the entire content of this entry.

The full treatment

First look: a clean transfer versus a messy merge

The billiard-ball case is close to what physicists call a perfectly elastic collision: the balls are hard, nearly undeformed, and separate cleanly after contact, with essentially none of the collision's energy diverted into heat, sound, or lasting deformation. The clay case is a perfectly inelastic collision: the two objects deform permanently, generate heat and sound in the process, and end up moving together as one lump, the maximum possible loss of relative motion consistent with momentum still being conserved. Nearly every real-world collision, a car crash, two hockey pucks, a bat striking a ball, sits somewhere between these two extremes, but the extremes are where the physics is cleanest and where you should build your intuition first.

Building the idea: one law that always holds, one that sometimes does

Momentum conservation, developed two entries back, rests only on Newton's third law: whatever force one colliding body exerts on the other during contact, the other exerts back with equal size and opposite direction, for the same duration of contact. That reasoning does not care whether the bodies are steel, rubber, or clay, so momentum, mass times velocity added up with direction, is conserved in absolutely every collision between two bodies with no external force acting during the brief contact. Kinetic energy conservation is a separate, stronger, and optional condition. It only holds if none of the kinetic energy present before the collision is diverted into some other form, permanent deformation, heat from internal friction as the materials compress and rebound imperfectly, or sound radiating away. Whether that diversion happens depends on the materials and geometry involved, not on any law of physics that must hold, which is why kinetic energy conservation needs to be checked, and momentum conservation does not.

The formal model: two masses, one dimension

Consider two masses, m1 moving at initial velocity u1 and m2 moving at initial velocity u2, colliding head on along a single line, ending at final velocities v1 and v2. Momentum conservation always gives one equation: m1 times u1, plus m2 times u2, equals m1 times v1, plus m2 times v2. If the collision is also perfectly elastic, kinetic energy conservation gives a second equation: one half m1 u1 squared, plus one half m2 u2 squared, equals one half m1 v1 squared, plus one half m2 v2 squared. Combining these two equations algebraically (the energy equation factors using the difference of squares, and dividing by the momentum equation cancels common factors) produces a strikingly simple result: u1 minus u2 equals minus of v1 minus v2. In words, the relative speed at which the two bodies approach each other before collision equals the relative speed at which they separate afterward, for a perfectly elastic collision. Apply this to the billiard-ball case, m1 equals m2, u2 equals zero: solving the two equations together gives v1 equals zero and v2 equals u1, the moving ball stops, the struck ball takes on exactly its speed, matching the observation that started this entry.

For a perfectly inelastic collision, the two bodies move together afterward, so v1 equals v2, call this common velocity v. Momentum conservation alone then gives v equals m1 u1 plus m2 u2, divided by the total mass m1 plus m2, no separate energy equation needed, because kinetic energy is not conserved here and asking it to balance would be asking the wrong question. The kinetic energy present afterward, one half times the total mass times v squared, is provably less than the kinetic energy present before whenever the two bodies had different velocities beforehand, and the missing amount has gone into heat, sound, and deformation.

The continuum between the extremes

Real collisions fall between these two clean cases, and the coefficient of restitution, the ratio of the relative separation speed to the relative approach speed, measures where on that continuum a given collision sits: a value of one recovers the perfectly elastic case, a value of zero recovers the perfectly inelastic case, and values in between describe everything from a bouncing rubber ball, closer to one, to a dropped bag of wet sand, closer to zero.

Lineage

Christiaan Huygens, John Wallis, and Christopher Wren worked out the rules governing colliding bodies in papers to the Royal Society between 1668 and 1669, correcting Descartes's earlier, direction-blind account of conserved motion and establishing much of the elastic-collision mathematics used here before Newton's Principia appeared in 1687. Newton himself verified these rules experimentally using pendulums of differing materials, recorded as corollaries following his Laws of Motion, an early instance of using a controlled swinging-collision apparatus, the ancestor of the familiar desktop toy sometimes called Newton's cradle, to test how much kinetic energy a given pair of colliding materials actually preserves. The reconciliation of "lost" kinetic energy in inelastic collisions with the broader principle of energy conservation, recognizing that the missing energy becomes heat rather than vanishing, came later, through the nineteenth-century work establishing the mechanical equivalent of heat.

The strongest case for it

The elastic-inelastic distinction, built on top of momentum conservation, does real explanatory and predictive work across scales. A ballistic pendulum measures a bullet's speed by treating the bullet's embedding in a block of wood as a perfectly inelastic collision, using momentum conservation to find the block's initial swing speed, then energy conservation on the swinging block itself. Particle physics experiments infer properties of subatomic particles from the momenta and energies of collision products, checking whether a given collision behaved elastically or not to identify what happened at the point of impact. Vehicle safety engineering treats a crash as a highly inelastic collision on purpose, since diverting kinetic energy into controlled deformation of the vehicle's structure, rather than into the passengers, is exactly what protects people inside.

The strongest case against it

Very few real collisions are purely one extreme or the other, and treating a real, partially elastic collision as if it were perfectly elastic overstates the final speeds and understates the energy actually lost, while assuming perfectly inelastic when objects actually rebound somewhat makes the opposite error. The one-dimensional formulas derived here also do not carry over directly to glancing, off-center collisions in two or three dimensions, which need the full vector form of momentum conservation plus a separate assumption, often the geometry of the impact, to pin down the outgoing directions. A common misconception is assuming momentum conservation and kinetic energy conservation are the same statement, or that one implies the other; they do not, momentum is conserved regardless of elasticity, kinetic energy is not, and confusing the two leads to setting up an energy equation for a collision, like two clumps of clay, where no such equation holds.

Where it stands now

The elastic and perfectly inelastic collision formulas are broad-consensus physics, unchanged since their seventeenth-century origin and resting on the even more secure foundation of momentum conservation. The coefficient of restitution remains the standard practical measure for characterizing real collisions that fall between the two idealized extremes.

Test yourself

Two train cars on a level, frictionless track, one moving and one at rest, are fitted with couplers that lock on contact, so they move off together after impact. Given both masses and the moving car's initial speed, find their common final speed, and compute exactly how much kinetic energy is lost in the coupling, stating where that energy has gone. Then suppose the same two cars instead have elastic bumpers that let them bounce apart cleanly instead of coupling; using the same masses and initial speed, find both final velocities, and explain, without redoing the momentum calculation, why the total momentum you get in this second scenario must be identical to the first even though the two final-velocity answers look completely different.