Collision Equations in Physics- Elastic and Inelastic Collisions

What Collision Equations Actually Tell You

In physics, a collision happens when two objects interact and exchange momentum and energy. That's it. No mystery here. The equations you'll learn aren't complicated—they just describe how momentum and energy behave when objects bump into each other.

Most students struggle with collision problems because they memorize formulas without understanding which quantities are conserved. Fix that first, and everything else clicks.

The Two Types of Collisions You Need to Know

Elastic Collisions

In an elastic collision, both momentum and kinetic energy are conserved. The objects bounce off each other without losing energy to deformation, heat, or sound.

Real-world examples are rare. Billiard balls come close. Gas molecules in ideal conditions behave elastically. That's about it.

Inelastic Collisions

In an inelastic collision, momentum is conserved but kinetic energy is not. Some energy converts to other forms—deformation, heat, sound.

Car crashes. Football tackles. Dropping a ball that bounces lower each time. These are all inelastic.

When objects stick together after colliding, that's a perfectly inelastic collision. It's the easiest type to solve because you combine the masses.

The Core Equations

Every collision problem starts with two equations. Memorize them:

Momentum Conservation

p = mv

Total momentum before = total momentum after

m₁v₁ + m₂v₂ = m₁v₁' + m₂v₂'

Variables with primes (') indicate values after the collision.

Kinetic Energy Conservation (Elastic Only)

KE = ½mv²

½m₁v₁² + ½m₂v₂² = ½m₁v₁'² + ½m₂v₂'²

You only use this equation for elastic collisions. For inelastic collisions, kinetic energy conservation doesn't apply—don't pretend it does.

Elastic vs Inelastic: Side by Side

How to Solve Any Collision Problem

Stop guessing. Follow this process:

Step 1: Identify the Type

Is kinetic energy conserved? If yes, elastic. If not, inelastic. If objects stick together, perfectly inelastic. This determines which equations you use.

Step 2: Write Down What You Know

List all given masses and velocities before the collision. Note which variables you're solving for.

Step 3: Apply Momentum Conservation

Set up: m₁v₁ + m₂v₂ = m₁v₁' + m₂v₂'

For perfectly inelastic: m₁v₁ + m₂v₂ = (m₁ + m₂)v'

Step 4: Apply Energy Equation (If Elastic)

Use the kinetic energy equation only if you've confirmed it's elastic.

Step 5: Solve the System

You have two equations and typically two unknowns. Solve algebraically or plug in numbers.

Working Example: Two Hockey Pucks

Problem: A 0.5 kg hockey puck moving at 10 m/s hits a stationary 0.3 kg puck. After the collision, the first puck slows to 4 m/s. What is the second puck's velocity?

Solution:

Given: m₁ = 0.5 kg, v₁ = 10 m/s, v₂ = 0, v₁' = 4 m/s, m₂ = 0.3 kg

Using momentum conservation:

(0.5)(10) + (0.3)(0) = (0.5)(4) + (0.3)(v₂')

5 = 2 + 0.3v₂'

v₂' = 3 / 0.3 = 10 m/s

The second puck flies off at 10 m/s. Notice total momentum stayed at 5 kg·m/s.

Working Example: Perfectly Inelastic Collision

Problem: A 2 kg cart moving at 5 m/s collides with a stationary 3 kg cart. They stick together. What is their combined velocity?

Solution:

m₁v₁ + m₂v₂ = (m₁ + m₂)v'

(2)(5) + (3)(0) = (5)v'

v' = 10/5 = 2 m/s

Simple. The combined mass moves slower because the same momentum is spread over more mass.

The Coefficient of Restitution

Sometimes you'll see e in collision problems. This is the coefficient of restitution—a number describing how "bouncy" a collision is.

• e = 1: Perfectly elastic
• 0 < e < 1: Partially inelastic
• e = 0: Perfectly inelastic (objects stick)

e = (v₂' - v₁') / (v₁ - v₂)

You won't always need this, but it shows up in advanced problems and exam questions.

Common Mistakes to Avoid

• Using kinetic energy conservation for inelastic collisions—it's wrong and will cost you points
• Forgetting that velocity is a vector—direction matters, use negative signs for opposite directions
• Confusing momentum (½mv²) with kinetic energy—different equations, different purposes
• Solving without identifying the collision type first—wastes time and leads to wrong equations

When to Use Each Equation

Most collision problems give you partial information. Here's your decision tree:

• Objects stick together → Use momentum equation with combined mass
• Problem states "elastic" → Use both momentum and kinetic energy equations
• Problem states "inelastic" → Use momentum equation only
• Given initial velocities, need final velocities → Solve the system of equations

Quick Reference Formulas

Two-body elastic collision (final velocities):

v₁' = [(m₁ - m₂)/(m₁ + m₂)]v₁ + [(2m₂)/(m₁ + m₂)]v₂

v₂' = [(2m₁)/(m₁ + m₂)]v₁ + [(m₂ - m₁)/(m₁ + m₂)]v₂

These come from solving the system of both conservation equations. You can derive them or memorize them—your call.

Perfectly inelastic (final velocity):

v' = (m₁v₁ + m₂v₂)/(m₁ + m₂)

This one is worth memorizing. It shows up constantly.

Bottom Line

Collision equations aren't hard. They're straightforward applications of conservation laws. The only thing that trips most people up is knowing which conservation law applies and when. Elastic means kinetic energy is conserved. Inelastic means it isn't. Momentum is always conserved—use it.