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Apply Physics to
Collision Design

Master the principles of momentum, impulse, and energy transfer to analyze and design collision systems in the real world.

High School Physics

What Is a Collision?

A collision occurs when two or more objects exert forces on each other over a short time interval. In physics, collisions are not just car crashes: they include billiard balls striking, molecules bouncing, and even a baseball bat hitting a ball.

Before A B Approaching After A B Separating

Momentum: The Foundation

Linear momentum is the product of an object's mass and velocity. It describes how hard it is to stop a moving object.

Mass \( m \)
Measured in kilograms (kg). More mass means more momentum at the same speed.
Velocity \( \vec{v} \)
Measured in m/s with direction. Doubling speed doubles momentum.
Momentum \( \vec{p} \)
A vector quantity. A truck at 10 m/s has far more momentum than a tennis ball at 50 m/s.

Impulse and Force Over Time

Impulse is the change in momentum of an object. It equals the average force multiplied by the time interval over which that force acts.

Time (s) Force (N) Impulse = Area
\( \Delta t \)
\( F_{\text{peak}} \)

Conservation of Momentum

In an isolated system (no net external forces), the total momentum before a collision equals the total momentum after.

Before Collision
\(2 \text{ kg}\)
\(3 \text{ m/s}\)
\(1 \text{ kg}\)
 at rest
\( p_{\text{total}} = 6 \text{ kg·m/s} \)
 After Collision
\(2 \text{ kg}\)
\(1 \text{ kg}\)
\(1 \text{ m/s}\)
\(4 \text{ m/s}\)
\( p_{\text{total}} = 6 \text{ kg·m/s} \)
 ✓ Momentum Conserved

Types of Collisions

Collisions are classified by what happens to kinetic energy. Momentum is conserved in all types, but kinetic energy may or may not be.

Elastic
Both momentum and kinetic energy are conserved. Objects bounce off each other. Example: billiard balls.
Inelastic
Momentum is conserved, but some kinetic energy is lost to heat, sound, or deformation. Most real collisions.
Perfectly Inelastic
Objects stick together after collision. Maximum kinetic energy loss. Example: car crash with entangled wreckage.

Elastic vs. Perfectly Inelastic

Elastic Perfectly Inelastic
Momentum Conserved Conserved
Kinetic Energy Conserved NOT conserved
After Collision Objects separate Objects stick together
Equations 2 equations (p + KE) 1 equation (p only)
Real Example Billiard balls, atomic nuclei Car pileup, clay balls

Kinetic Energy Transfer

The kinetic energy of a moving object is given by:

In inelastic collisions, the "lost" kinetic energy is not destroyed: it transforms into thermal energy, sound, and deformation of materials. This energy transformation is exactly what collision design harnesses.

Energy Distribution (Inelastic Collision) 100% KE Before
\(50\% \text{ KE}\)
 Heat Sound Deformation After

How to Solve Collision Problems

1
Identify the system. Which objects are involved? Is the system isolated (no external net force)?
2
Classify the collision. Elastic, inelastic, or perfectly inelastic? Do the objects stick together?
3
Write conservation of momentum. \( m_1 v_{1i} + m_2 v_{2i} = m_1 v_{1f} + m_2 v_{2f} \)
4
If elastic, add KE conservation. \( \frac{1}{2}m_1 v_{1i}^2 + \frac{1}{2}m_2 v_{2i}^2 = \frac{1}{2}m_1 v_{1f}^2 + \frac{1}{2}m_2 v_{2f}^2 \)
5
Solve for unknowns. Use algebra. For perfectly inelastic: \( v_f = \frac{m_1 v_{1i} + m_2 v_{2i}}{m_1 + m_2} \)
6
Check your answer. Does momentum balance? Are signs (directions) correct?

Example 1: Perfectly Inelastic

A 4 kg cart moving at 6 m/s collides with a 2 kg cart at rest. They stick together. Find the final velocity.

Before
\(4 \text{ kg}\)
\(6 \text{ m/s}\)
\(2 \text{ kg}\)
 at rest After (stuck)
\(6 \text{ kg}\)
\((4+2)\)
\(? \text{ m/s}\)

Example 2: Elastic Collision

A 3 kg ball moving at 8 m/s hits a 3 kg ball at rest in a perfectly elastic collision. What happens?

1
Momentum: \( (3)(8) + (3)(0) = 3 v_{1f} + 3 v_{2f} \) so \( v_{1f} + v_{2f} = 8 \)
2
Kinetic Energy: \( \frac{1}{2}(3)(64) = \frac{1}{2}(3) v_{1f}^2 + \frac{1}{2}(3) v_{2f}^2 \) so \( v_{1f}^2 + v_{2f}^2 = 64 \)
3
Result: \( v_{1f} = \) \( 0 \text{ m/s} \), \( v_{2f} = \) \( 8 \text{ m/s} \). Complete transfer of motion.

Example 3: Crumple Zone Design

A 1500 kg car traveling at 20 m/s crashes into a wall. Compare the force on the driver for two scenarios:

Rigid Frame
\( \Delta t = 0.05 \text{ s} \)
\( F = \frac{1500 \times 20}{0.05} = 600,000 \text{ N} \)
Crumple Zone
\( \Delta t = 0.5 \text{ s} \)
\( F = \frac{1500 \times 20}{0.5} = 60,000 \text{ N} \)

Coefficient of Restitution

The coefficient of restitution \( e \) measures how "bouncy" a collision is: the ratio of relative speeds after to before the collision.

0 Stick 0.5 Partially Bouncy 1 Perfect Bounce Coefficient of Restitution (e)

Real-World Design Applications

Airbags
Increase \( \Delta t \) for the occupant, reducing peak force. The bag deflates slowly, absorbing energy over ~0.1 s.
Crumple Zones
Car body deforms plastically, converting kinetic energy to deformation work and extending collision time.
Helmets
Foam liner crushes on impact, spreading force over time and area. Designed to be inelastic (low \( e \)) to absorb energy in a single large impact.

Common Mistakes

✕ Forgetting Direction
Momentum is a vector. If objects move in opposite directions, one velocity must be negative. Ignoring signs gives wrong answers.
✕ Applying KE Conservation to Inelastic
Kinetic energy is only conserved in elastic collisions. Using \( KE_i = KE_f \) for inelastic collisions will give incorrect results.
✕ Confusing Force and Impulse
Impulse \( (F \cdot \Delta t) \) is the change in momentum, not the force itself. A small force over a long time can produce the same impulse as a large force over a short time.

Summary of Key Equations

Concept Equation When to Use
Momentum \( \vec{p} = m\vec{v} \) Always
Impulse \( \vec{J} = F_{\text{avg}} \Delta t = \Delta \vec{p} \) Force/time problems
Conservation \( \Sigma \vec{p}_i = \Sigma \vec{p}_f \) All isolated collisions
Perfectly Inelastic \( v_f = \frac{m_1 v_{1i} + m_2 v_{2i}}{m_1 + m_2} \) Objects stick together
Kinetic Energy \( KE = \frac{1}{2}mv^2 \) Elastic collisions, energy analysis
Restitution \( e = \frac{|v_{2f} - v_{1f}|}{|v_{1i} - v_{2i}|} \) Classifying bounce

Question 1 of 5

A 5 kg object moving at 10 m/s collides with a 5 kg object at rest and they stick together. What is the final velocity?

Question 2 of 5

Which quantity is conserved in all collisions (assuming an isolated system)?

Question 3 of 5

An airbag reduces injury by:

Question 4 of 5

Two objects of equal mass collide elastically. Object A is moving and object B is at rest. After the collision:

Question 5 of 5

A collision has a coefficient of restitution \( e = 0 \). This means: