1D Collision – Elastic / Perfectly Inelastic – Tutorial

What Is a 1D Collision?

A 1D collision is one where two objects move along the same straight line (single axis) and the collision is analysed along that axis. This model is a simplified but powerful representation of real-world collisions: cars crashing on a straight road, air-hockey pucks, wagons on a track, low-friction systems…

"1D" means we distinguish directions only as positive (+) and negative (−): for example, right is + and left is −. The choice is arbitrary; what matters is staying consistent.

What Is Conserved and What Changes?

The collision happens in a very short time, during which the objects exert very large forces on each other. However, (if there is no external push or pull) the total momentum of the system is conserved. Whether kinetic energy is conserved depends on the type of collision.

1) Momentum (Conserved in both types)

Momentum is the "carried quantity" of motion:

Total momentum for two objects:

Conservation of momentum means:

Why is it conserved? Because internal collision forces are equal and opposite (Newton's 3rd law), so if no external net impulse acts on the system, total momentum does not change.

2) Kinetic Energy (Conserved only in elastic collisions)

Kinetic energy:

Total kinetic energy:

An elastic collision ideally means "total kinetic energy before = total kinetic energy after". In a perfectly inelastic collision, some kinetic energy is converted to other forms: heat, sound, permanent deformation, microscopic vibrations…

Collision Types

A) Perfectly Inelastic (Stick Together)

In this scenario the objects "stick together" after the collision and move as a single object at the same velocity: v₁' = v₂' = v'. Momentum is still conserved but kinetic energy decreases.

Momentum equation:

Common velocity:

B) Elastic Collision

In an elastic collision two things are simultaneously true:

• Momentum is conserved
• Kinetic energy is conserved

Combining these two conditions gives the closed-form final velocity formulas for 1D:

Building Intuition: Special Cases

1) m₁ = m₂ (equal masses)

In an elastic collision the velocities effectively "swap":

• v₁' = v₂
• v₂' = v₁

This is visibly observed in examples like billiard balls.

2) m₁ ≫ m₂ (object 1 much heavier)

The heavy object's velocity barely changes while the light one can bounce back. This intuitive result follows from the formulas: (m₁+m₂) ≈ m₁ dominates the denominator.

What Does Energy Loss Mean? (Interpreting ΔE)

• If you selected Elastic, ideally ΔE ≈ 0 (small differences may come from rounding).
• If you selected Perfectly inelastic, ΔE is typically negative (kinetic energy decreased).

This does not mean energy disappeared; total energy is conserved but kinetic energy converts to other forms: heat, sound waves, permanent deformation, microscopic vibrations from surface friction…

Important Notes (Model Limitations)

• This model is one-dimensional; real collisions are often 2D or 3D.
• External forces and friction are neglected; for long-contact scenarios this assumption may break down.
• The "elastic" idealisation is only approximate in practice (like billiard balls), not perfectly exact.
• "Perfectly inelastic" is the extreme case; most real collisions fall somewhere in between.

Tip: A natural next step would be to add a coefficient of restitution (e) to support partially elastic collisions (0<e<1).