How to Solve an Elastic or Inelastic Collision

Two vehicles collide and stick together. Momentum is conserved, kinetic energy is not, and 270 kJ has gone somewhere.

Where it goes is the entire point of vehicle safety design, and the arithmetic that finds it is the same arithmetic that solves the collision.

Momentum is always conserved

In an isolated system, total momentum before equals total momentum after. Always, for every collision, regardless of type.

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

That single equation is what makes collisions solvable. It holds whether the objects bounce apart, stick together, or deform.

Signs carry direction. A 1500 kg vehicle at +20 m/s and a 1000 kg vehicle at −10 m/s have total momentum of 30,000 − 10,000 = 20,000 kg m/s, and that figure is unchanged afterwards.

Kinetic energy is the variable

What distinguishes collision types is what happens to kinetic energy.

Perfectly elastic — kinetic energy conserved. Objects bounce apart with no energy lost to deformation or heat. A close approximation for billiard balls and gas molecules; nothing macroscopic is exactly elastic.

Perfectly inelastic — objects stick together and move as one. This loses the maximum kinetic energy consistent with conserving momentum, which is the key phrase: momentum constrains how much can be lost, and inelastic collisions lose all of it.

Real collisions — somewhere between, described by a coefficient of restitution between 0 and 1. It is measured for a given pair of materials rather than derived.

Where the lost energy goes, and why it matters

Deformation, heat and sound. Predominantly deformation.

That is the design principle behind vehicle crumple zones, and it inverts the intuition that a stronger car is a safer one. A rigid vehicle transfers the collision’s energy into rapid deceleration of the occupants; a vehicle that deforms absorbs energy over a longer distance and time, reducing the peak force on the people inside.

The physics: force depends on the rate of momentum change. Extending the collision from 50 milliseconds to 150 reduces the peak force by roughly a factor of three, and that difference is the difference between injuries.

So a car destroying itself in a crash is doing exactly what it was designed to do. The passenger cell stays rigid; everything in front of it is meant to fold.

Two dimensions need vectors

Everything above assumes a head-on collision along one line. Real collisions are usually oblique, and the extension is straightforward with one addition.

Momentum is a vector, so it is conserved independently in each direction. Resolve the initial velocities into x and y components, apply conservation separately to each, and recombine.

That gives two equations, which is not enough to solve a two-dimensional collision on its own — a third piece of information is needed, typically the deflection angle of one object or the assumption that the collision is elastic.

The practical consequence in accident reconstruction is that a two-dimensional collision cannot be solved from masses and initial velocities alone. Something else must be measured, usually the final direction of travel from skid marks or resting positions.

Reference frames simplify the algebra

A technique worth knowing for elastic collisions specifically.

In the centre of mass frame — the frame moving with the system’s overall momentum — the total momentum is zero by construction. That makes an elastic collision almost trivial: each object simply reverses its velocity in that frame.

Transforming into the frame, reversing, and transforming back gives the answer with considerably less algebra than solving the simultaneous momentum and energy equations directly.

It also makes the physics clearer. The reason a light object bounces off a heavy one at nearly its original speed, while a heavy object barely deflects, is obvious in the centre of mass frame and buried in the algebra otherwise.

Common mistakes to avoid

• Assuming kinetic energy is conserved, which is true only for perfectly elastic collisions.
• Dropping the signs and treating velocities as speeds, which breaks the momentum sum.
• Using a coefficient of restitution from a different material pair.
• Applying one-dimensional equations to an oblique collision, which needs vector components.
• Concluding that a car that deformed badly performed badly.

How to do it with Momentum Collision Calculator

The Momentum Collision Calculator solves both cases.

1. Enter both masses and velocities, with signs for direction.
2. Choose the collision type, or supply a coefficient of restitution.
3. Read the final velocities and check momentum is conserved.
4. Read the kinetic energy loss — that is the physically interesting number.

Other physics tools are in the tools directory.

Frequently asked questions

Is momentum always conserved?

In an isolated system, yes, in every collision. Kinetic energy is only conserved in a perfectly elastic collision, which is why the two are treated separately.

What is a coefficient of restitution?

A number from 0 to 1 describing how elastic a collision is — zero for objects that stick together, one for perfectly elastic. It is measured for a material pair rather than calculated.

Where does the lost energy go?

Deformation, heat and sound, mostly deformation. In vehicle design that is deliberate: material that folds absorbs energy that would otherwise reach the occupants.

Final thought

Check momentum is conserved in your answer, then look at the energy. The first confirms the arithmetic; the second is what the collision actually did.

Try the free Momentum Collision Calculator