Collisions

A collision is a short interaction during which objects push strongly on one another and change their motion. The details of the contact can be messy, but the bookkeeping is not: in an isolated system, total momentum is conserved. The big question is what happens to the kinetic energy.

Momentum

Momentum quantifies how difficult it is to stop or redirect motion. For a single object,

p⃗=mv⃗.\vec{p} = m\vec{v}.

For a system of objects, we add the momenta of every part:

p⃗total=∑imiv⃗i.\vec{p}_{\text{total}} = \sum_i m_i \vec{v}_i.

If external forces are negligible during the collision, then the system is isolated and

p⃗before=p⃗after.\vec{p}_{\text{before}} = \vec{p}_{\text{after}}.

During collisions, objects may bounce apart, deform, or stick together. Although individual momenta can change dramatically during impact, the vector sum of the system’s total momentum stays fixed throughout.

Energy

Kinetic energy is the energy of motion:

K=12mv2.K = \frac{1}{2}mv^2.

Total energy is always conserved, but kinetic energy is not always conserved by itself. During a collision, some of the original kinetic energy can become

  • thermal energy from friction and internal vibration.
  • sound produced during impact.
  • permanent deformation of the objects.

So when we classify collisions, we are really asking whether the kinetic energy remains kinetic after the interaction.

Elastic Collisions

An elastic collision conserves both total momentum and total kinetic energy:

p⃗before=p⃗after,Kbefore=Kafter.\vec{p}_{\text{before}} = \vec{p}_{\text{after}}, \qquad K_{\text{before}} = K_{\text{after}}.

These collisions are good models for idealized billiard balls, gas particles, and many microscopic scattering processes. In an elastic collision, the objects separate after the collision, and no kinetic energy is “lost” to heat or deformation.

Inelastic Collisions

An inelastic collision still conserves momentum, but it does not conserve kinetic energy:

p⃗before=p⃗after,Kafter<Kbefore.\vec{p}_{\text{before}} = \vec{p}_{\text{after}}, \qquad K_{\text{after}} < K_{\text{before}}.

The “missing” kinetic energy has not vanished. It has been transformed into other forms of energy inside or around the colliding objects. A special case is the perfectly inelastic collision, where the objects stick together and move with one shared final velocity. In one dimension,

m1v1,i+m2v2,i=(m1+m2)vf.m_1 v_{1,i} + m_2 v_{2,i} = (m_1 + m_2) v_f.

1D Collisions

Explore the differences between elastic and inelastic collisions.

Collision Guide

Track what is conserved and what changes form.

In an isolated system, momentum is conserved in every collision. The key distinction is whether kinetic energy remains kinetic or is redistributed into other forms of energy.

What to Track

Elastic

Momentum and kinetic energy are both conserved.

Inelastic

Momentum is conserved, but kinetic energy is not.

Momentum

Total momentum stays constant because the interaction forces are internal to the system.
Total momentum still stays constant, even when the objects crumple, stick, or warm up.

Kinetic Energy

Kinetic energy before and after the collision is the same.
Some kinetic energy is transformed into sound, thermal energy, or deformation.

After the Impact

Objects separate with new speeds that still satisfy both conservation laws.
Objects may separate more slowly, or in the perfectly inelastic case move together.

Always True

Total momentum: pbefore = pafter

As long as outside forces are negligible, internal pushes and pulls cancel in pairs.

Perfectly Inelastic Limit

m1v1,i + m2v2,i = (m1 + m2)vf

The objects share a common final speed, which tells you the lost kinetic energy was converted into other forms.

What to Notice

As you compare different collision types, pay attention to three patterns:

  • Momentum conserved in both elastic and inelastic processes.
  • Kinetic energy is the classifier: if it is conserved, the collision is elastic; if not, it is inelastic.
  • “Lost” kinetic energy is really transformed energy, not destroyed energy.

Collisions Checkpoint

Question 1 of 3

Two carts collide and stick together. Which statement must still be true if outside forces are negligible during the collision?

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