Inelastic Collision

An inelastic collision is a collision during which the total kinetic energy of the system decreases, converted into deformation, heat and, to a small extent, sound. Momentum, on the other hand, is conserved, as in any collision between bodies that are isolated during the contact.

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How to measure it in class

The quantity to measure is the coefficient of restitution e, which quantifies how much of the motion survives the collision. The simplest approach is to study a ball bouncing on the floor: this is an inelastic collision between the ball and the Earth, and each bounce provides a measurement.

Steps:

Place the smartphone on the floor, a few tens of centimeters from the impact point, and start the audio recording; move it further away if the microphone saturates, because a clipped peak is hard to time
Drop the ball with no initial velocity from a measured height h₀, and let it bounce five or six times
Locate the instants of the successive impacts on the recording: they are sharp peaks, easy to time
Calculate the flight times between two consecutive impacts, t₁, t₂, t₃…
Deduce the coefficient of restitution bounce by bounce: e = t(n+1) / t(n)
Calculate the fraction of energy lost at each bounce: 1 − e², and check whether e stays constant from one bounce to the next
Stop the analysis before the last bounces, which are too close together to be distinguished, then start again on a different floor and compare

Scientific activities on this topic

Coefficient of restitution of a bouncing ball - measure e from the impact times and quantify the energy lost at each bounce
Elastic collision: billiards - the opposite reference case, to compare: a quasi-elastic collision, where e stays close to 1

Learn more

What is conserved and what is not

During the few milliseconds of contact, the interaction forces between the two bodies are enormous compared with their weight and with friction. The two-body system is therefore quasi-isolated, and its total momentum is the same before and after. Kinetic energy, however, decreases. There is no violation of the principle of conservation of energy here: total energy is always conserved, it simply changes form. It goes from an ordered overall motion to a disordered agitation of atoms, which cannot be recovered.

Where the lost energy goes

Three destinations, very unequal. Plastic deformation first: bending a metal sheet, squashing modeling clay, this costs work that is never returned. Heat next, which is by far the main share; the temperature rise is real but often too small to be felt, only a few tenths of a degree. Sound finally: the noise of the impact is indeed lost kinetic energy, but it represents a tiny share, well below one percent. We hear it because the ear is extraordinarily sensitive, not because the energy involved is large.

The coefficient of restitution

Newton introduced this number e, the ratio of the relative separation speed to the relative approach speed. It varies between 0 and 1 and serves as a single dial between the two extremes: e = 1 for an elastic collision, where nothing is lost; e = 0 for a perfectly soft collision, where the bodies remain stuck together. In between, the collision is simply said to be inelastic. The fraction of kinetic energy lost equals 1 − e²: with e = 0.8, already 36% of the energy disappears at each collision. Some orders of magnitude: steel on steel 0.90 to 0.95; basketball on parquet 0.75 to 0.80; tennis ball on a hard floor about 0.75; golf ball on concrete 0.85; modeling clay, sand, soft clay, close to 0. These values always characterize a pair of materials, never a single object: the same ball gives a different e on tiles and on parquet.

The perfectly soft collision is the maximum loss

It is sometimes believed that a collision can dissipate all the kinetic energy. This is false in general: the momentum to be conserved forces the system to keep at least the kinetic energy of its center of mass. The only case where everything can disappear: two bodies with opposite momenta, like two identical cars colliding head-on at the same speed. They stop dead, and all of the kinetic energy goes into deformation and heat.

Why inelasticity protects

In an accident, the energy must be dissipated somewhere. The crumple zones of a car are deliberately very inelastic: they crush instead of bouncing. A rebound would be far more dangerous, because it would imply a larger change in velocity for the occupants, and therefore a higher force. Crushing the structure also lengthens the duration of the impact, which further reduces the force experienced.

Formula

Conservation of momentum, valid in all collisions:

m₁v₁ + m₂v₂ = m₁v₁′ + m₂v₂′

Kinetic energy in an inelastic collision:

Ek(after) < Ek(before)

Common velocity after a perfectly soft collision (e = 0):

v′ = (m₁v₁ + m₂v₂) / (m₁ + m₂)

Energy dissipated in a perfectly soft collision:

ΔEk = ½ × [m₁m₂ / (m₁ + m₂)] × (v₁ − v₂)²

Coefficient of restitution and fraction of energy lost:

e = |v₂′ − v₁′| / |v₁ − v₂| ; fraction lost = 1 − e²

Coefficient of restitution measured on a bounce, from heights or flight times:

e = √(h(n+1) / h(n)) = t(n+1) / t(n)

where:

m₁, m₂: masses of the two bodies (kg)
v₁, v₂: velocities before the collision, signed values along an oriented axis (m·s⁻¹)
v₁′, v₂′: velocities after the collision (m·s⁻¹)
v′: common velocity after a soft collision (m·s⁻¹)
ΔEk: kinetic energy dissipated (J)
e: coefficient of restitution, between 0 and 1 (dimensionless)
h(n), t(n): height and flight time of the n-th bounce (m, s)

Application examples

Tennis ball dropped from 2.54 m. With e ≈ 0.75, it rises back to about 1.43 m, that is e² ≈ 56% of the initial height: 44% of the energy has disappeared after the first bounce. This rebound height is in fact standardized by the international federation.
Car against a wall. A 1,200 kg car at 50 km/h, that is 13.9 m·s⁻¹, carries 116 kJ. It stops without bouncing: almost all of these 116 kJ go into deforming the bodywork. The momentum has not disappeared - it has been transferred to the wall and to the Earth, whose velocity changes in a totally imperceptible way.
Two railway cars coupling. A 20 t car rolling at 2 m·s⁻¹ hooks onto an identical one at rest: the pair moves off at 1 m·s⁻¹. Kinetic energy drops from 40 kJ to 20 kJ; half is dissipated in the buffers. A textbook perfectly soft collision.
Ballistic pendulum. A rifle bullet embeds itself in a suspended wooden block: a perfectly soft collision. By measuring the height the block rises, you can work back to the bullet’s speed. The calculation only works if you use conservation of momentum during the collision, then conservation of mechanical energy afterwards - never energy during the collision.
Fall of a meteorite. The impact is extremely inelastic: the kinetic energy is converted into fracturing, heat and seismic waves that a smartphone accelerometer can detect at a distance.
Soft clay and modeling clay. A ball that squashes without bouncing has e ≈ 0: it is the ideal material for showing in a lab session that kinetic energy is not a conserved quantity.

FAQ

Q: If kinetic energy is not conserved, is the principle of conservation of energy violated? A: No, never. Total energy is conserved: what disappears as kinetic energy reappears as internal energy, that is, as heat and deformation of the material. It is just that this energy has become disordered, and it can no longer be fully converted back into motion.

Q: What is the difference between an inelastic collision and a perfectly soft collision? A: Any non-elastic collision is inelastic, meaning 0 ≤ e < 1. The perfectly soft collision is the special case e = 0, where the two bodies move off together at the same velocity. It is the one that dissipates the maximum energy compatible with conservation of momentum.

Q: Why is momentum conserved even though the bodies deform? A: Because the forces they exert on each other are opposite, according to Newton’s third law. What one loses, the other gains exactly. Deformation costs energy, but it neither creates nor destroys momentum.

Q: Can a collision dissipate 100% of the kinetic energy? A: Only if the total momentum of the system is zero before the collision, for example two identical bodies colliding head-on at the same speed. In all other cases, the system must keep moving to conserve p, and therefore retains some kinetic energy.

Q: Does the noise of the impact represent a lot of lost energy? A: Very little, generally well under 1%. Most of it goes into heat and deformation. Sound is an excellent marker of inelasticity - a perfectly elastic collision would be silent - but a very poor indicator of the amount of energy involved.

Elastic Collision - Momentum - Kinetic Energy - Conservation of Energy - Coefficient of Restitution - Mechanical Energy - Newton’s Third Law