Collision Equations- Physics Explained

What Are Collision Equations?

Collision equations describe what happens when two objects hit each other. That's it. They tell you how fast things are moving before and after impact, based on two non-negotiable laws: conservation of momentum and conservation of energy.

Physics doesn't care about your feelings about the outcome. The objects crash, momentum transfers, energy changes form, and the math tells you exactly where everything ends up.

The Two Types of Collisions You Need to Know

Elastic Collisions

In an elastic collision, objects bounce off each other like billiard balls. Kinetic energy is conserved — no energy gets lost to heat, sound, or deformation.

Real-world examples are rare. Supermarket checkout lanes and pool tables come close, but perfect elasticity doesn't exist in everyday life. Space is where you find truly elastic collisions.

Inelastic Collisions

Objects stick together or deform. Kinetic energy is NOT conserved — some of it converts to heat, sound, or deformation energy.

Car crashes are the textbook example. The crumpled metal and noise mean energy went somewhere other than motion.

Perfectly Inelastic Collisions

This is a subset of inelastic collisions where objects stick together after impact. They move as one mass afterward. Think of two railway cars coupling together.

The Core Equations

Here are the formulas that govern every collision problem you'll encounter:

Momentum Conservation (Always Applies)

This works for every collision, no exceptions:

m₁v₁ + m₂v₂ = m₁v₁' + m₂v₂'

m = mass, v = initial velocity, v' = final velocity. The total momentum before equals total momentum after. Always.

Energy Equation for Elastic Collisions

Add this when kinetic energy is conserved:

½m₁v₁² + ½m₂v₂² = ½m₁v₁'² + ½m₂v₂'²

Velocity Formulas for 1D Elastic Collisions

These are derived from the two conservation laws combined:

v₁' = [(m₁-m₂)v₁ + 2m₂v₂] / (m₁+m₂)

v₂' = [(m₂-m₁)v₂ + 2m₁v₁] / (m₁+m₂)

Use these when you need to find final velocities directly.

Comparison Table: Collision Types

Notice momentum is always conserved. That's not negotiable. Energy conservation depends on the collision type.

Getting Started: How to Solve Collision Problems

Here's the process. Follow it every time:

1. Identify your knowns — masses, initial velocities. Write them down.
2. Determine the collision type — elastic, inelastic, or perfectly inelastic.
3. Pick your equations — momentum equation is always first. Add energy equation only for elastic.
4. Set up your equations — substitute your known values.
5. Solve algebraically — isolate your unknown variable.
6. Check your work — verify momentum is conserved in your answer.

Example Problem

A 2 kg ball moving at 4 m/s hits a stationary 3 kg ball. Find their final velocities if the collision is perfectly elastic.

Step 1: m₁ = 2 kg, v₁ = 4 m/s, m₂ = 3 kg, v₂ = 0

Step 2: Elastic collision — use both momentum and energy equations

Step 3: Using the velocity formulas:

v₁' = [(2-3)(4) + 2(3)(0)] / (2+3) = [-4 + 0] / 5 = -0.8 m/s

v₂' = [(3-2)(0) + 2(2)(4)] / (2+3) = [0 + 16] / 5 = 3.2 m/s

The first ball bounces backward. The second ball takes most of the momentum and flies off at 3.2 m/s.

Common Mistakes to Avoid

• Using the energy equation for inelastic collisions — kinetic energy isn't conserved there
• Forgetting that velocity is a vector — direction matters in 1D problems
• Mixing up initial and final velocities in your equations
• Not converting units — mass in kg, velocity in m/s, or your answers will be wrong

When to Use Each Equation

If you only have momentum conservation, you have one equation. You can solve for one unknown.

If you have both momentum and energy conservation (elastic collision), you have two equations. You can solve for two unknowns.

If objects stick together (perfectly inelastic), combine the masses first, then apply momentum conservation. You get one equation for one unknown.