Collisions & Conservation Laws
Understanding momentum and energy conservation in one-dimensional collisions.
The Two Conservation Laws
In any collision, two fundamental quantities are conserved: momentum and (sometimes) kinetic energy. Understanding which is conserved determines the outcome of the collision.
🔄 Always Conserved
Momentum is conserved in all collisions (assuming no external forces).
⚡ Sometimes Conserved
Kinetic Energy is only conserved in elastic collisions.
Types of Collisions
Elastic (e = 1)
Both momentum and kinetic energy are conserved. Think of billiard balls or ideal gas molecules.
v₁' = [(m₁-m₂)v₁ + 2m₂v₂] / (m₁+m₂)
v₂' = [2m₁v₁ + (m₂-m₁)v₂] / (m₁+m₂)
Special case (equal masses): The bodies simply exchange velocities!
Perfectly Inelastic (e = 0)
Momentum is conserved but maximum kinetic energy is lost. The objects stick together after collision.
v' = (m₁v₁ + m₂v₂) / (m₁ + m₂)
All kinetic energy relative to the center of mass is converted to heat/sound.
Partially Elastic (0 < e < 1)
The most common real-world case. Some kinetic energy is lost.
Coefficient of Restitution (e)
e = (v₂' - v₁') / (v₁ - v₂)
- e = 1: Elastic
- e = 0: Perfectly inelastic
- 0 < e < 1: Partially elastic
Worked Example: Elastic Collision
Problem: A 2 kg ball moving at 4 m/s collides elastically with a stationary 2 kg ball. Find the final velocities.
Step 1: Conservation of Momentum
(2)(4) + (2)(0) = 2v₁' + 2v₂'
8 = 2v₁' + 2v₂'
Step 2: Relative Velocity (Elastic)
4 - 0 = -(v₁' - v₂')
v₂' - v₁' = 4
Step 3: Solve
From the two equations: v₁' = 0 m/s and v₂' = 4 m/s
✓ The balls exchange velocities!
Ready to Practice?
Test your understanding with randomized collision problems.
Start Practicing →Interactive Demo
Experiment with different masses, velocities, and collision types. Try setting equal masses in an elastic collision to see velocity exchange!