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Optimize Designs for
Collision Safety

Learn how impulse, momentum, and force govern collisions, and discover engineering strategies to reduce impact forces and save lives.

High School Physics

What Is a Collision?

A collision occurs when two objects exert forces on each other over a brief time interval. The key physics happens during this short interaction: momentum transfers, objects deform, and forces spike to extreme values.

Momentum
The product of mass and velocity: \( p = mv \). A moving object's "quantity of motion."
Impulse
The change in momentum: \( J = \Delta p = F_{\text{avg}} \cdot \Delta t \). Force multiplied by contact time.
Impact Force
The average force during collision: \( F_{\text{avg}} = \frac{\Delta p}{\Delta t} \). Shorter time means larger force.

The Impulse-Momentum Theorem

The impulse-momentum theorem is the foundation of collision safety design. It states that the impulse applied to an object equals the change in its momentum:

This means that for a given change in momentum \( m \cdot \Delta v \), the average force and the collision time are inversely related. If you increase \( \Delta t \), you decrease \( F_{\text{avg}} \).

Visualizing Force vs. Time

Time (s) Force (N)
High \(F\)
Short \( \Delta t \)
Low \(F\)
Long \( \Delta t \)
Area = \(J\)
Area = \(J\)
 Same impulse, different forces

From \( F = ma \) to Collision Design

Newton's Second Law can be rewritten using momentum. Starting from \( F = ma \) and noting \( a = \frac{\Delta v}{\Delta t} \):

1
Start: \( F = ma = m \cdot \frac{\Delta v}{\Delta t} \)
2
Rearrange: \( F \cdot \Delta t = m \cdot \Delta v = \Delta p \)
3
Solve for force: \( F_{\text{avg}} = \frac{\Delta p}{\Delta t} \)
4
Design insight: To make \( F \) smaller, make \( \Delta t \) larger while \( \Delta p \) stays constant.

Elastic vs. Inelastic Collisions

Elastic Inelastic
Momentum Conserved Conserved
Kinetic Energy Conserved Not conserved
Deformation Objects bounce back Objects crumple or stick
Energy goes to... Stays as KE Heat, sound, deformation
Safety relevance Increases magnitude of momentum change, \( |\Delta p| \) Crumpling absorbs energy

Real-World Safety Features

Crumple Zones
Metal structures at the front and rear of a car that collapse progressively, extending \( \Delta t \) from ~5 ms to ~100 ms.
Airbags
Inflate in ~30 ms and deflate slowly as the occupant presses into them, spreading deceleration over a longer time.
Seatbelts
Restrain the body so it decelerates with the car (not the dashboard). Force limiters allow slight belt extension to increase \( \Delta t \).
Helmets
A crushable foam liner absorbs impact energy and extends collision time from <1 ms (bare skull) to ~6 ms.

How a Crumple Zone Works

BEFORE Crumple Cabin
\(v\)
 AFTER Cabin intact
\(v = 0\)
 Absorbs energy Crushed Passenger safe

How to Solve Collision Safety Problems

1
Identify the change in momentum: Calculate \( \Delta p = m \cdot \Delta v \). Remember \( \Delta v = v_f - v_i \).
2
Determine the collision time: Find \( \Delta t \) for the scenario (with and without the safety feature).
3
Calculate the average force: Use \( F_{\text{avg}} = \frac{|\Delta p|}{\Delta t} \). Compare forces for different \( \Delta t \) values.
4
Evaluate the design: A larger \( \Delta t \) means a smaller force. Quantify the percentage reduction in force.
5
Check units: Mass in kg, velocity in m/s, time in s, force in Newtons.

Example 1: Car Hitting a Wall

A 1500 kg car traveling at 20 m/s hits a wall and comes to rest. Without a crumple zone, the collision takes 0.05 s. With a crumple zone, it takes 0.5 s. Find the force in each case.

1
Momentum change: \( \Delta p = m \cdot \Delta v = 1500 \times (0 - 20) = -30{,}000 \;\text{kg\cdot m/s} \)
2
Without crumple zone: \( F_{\text{avg}} = \frac{|\Delta p|}{\Delta t} = \frac{30{,}000}{0.05} = \) 600,000 N
3
With crumple zone: \( F_{\text{avg}} = \frac{|\Delta p|}{\Delta t} = \frac{30{,}000}{0.5} = \) 60,000 N

Example 2: Helmet Impact

A cyclist's head (mass 5 kg) hits the ground at 8 m/s. Without a helmet, the collision lasts 0.002 s. With a helmet, it lasts 0.012 s.

Without Helmet
\( \Delta p = 5(0-8) = -40 \;\text{kg\cdot m/s} \)
\( F_{\text{avg}} = \frac{|-40|}{0.002} = \) 20,000 N
With Helmet
\( \Delta p = -40 \;\text{kg\cdot m/s} \) (same)
\( F_{\text{avg}} = \frac{|-40|}{0.012} \approx \) 3,333 N

Example 3: Airbag + Seatbelt System

A 70 kg driver in a car moving at 30 m/s (about 67 mph) crashes. Compare three scenarios:

Scenario \( \Delta t \) (s) Force (N) Reduction
Dashboard (no restraint) 0.02 105,000 Baseline
Seatbelt only 0.15 14,000 87%
Seatbelt + Airbag 0.30 7,000 93%

The Energy Perspective

Impulse tells us about force reduction. But energy tells us where the kinetic energy goes. A car at 30 m/s has kinetic energy:

In a crash, this energy must go somewhere. Good design ensures it goes into controlled deformation, not into the passengers:

Crumple Zone
Absorbs ~70% of the KE through metal deformation
Airbag
Absorbs ~15% through gas compression and venting
Seatbelt
Absorbs ~10% through belt stretching and spooling

Design Optimization Principles

1
Maximize collision time (\( \Delta t \)): Use deformable materials, padding, and crumple structures.
2
Absorb kinetic energy: Convert KE into heat, sound, and permanent deformation rather than bounce-back.
3
Distribute force over area: Spread contact forces across a wide surface (e.g., airbag vs. steering column).
4
Redirect occupant motion: Keep people moving in a controlled path (seatbelts, side curtains) to avoid secondary impacts.

Common Mistakes

✕ Confusing Impulse with Force
Impulse \( J = F_{\text{avg}} \cdot \Delta t \) is not force. A larger impulse doesn't mean a larger force: it depends on how long the force acts.
✕ Ignoring Direction
Momentum is a vector. A car going from +20 m/s to 0 has \( \Delta v = -20 \;\text{m/s} \). Forgetting the sign gives wrong force direction.
✕ "Stronger = Safer"
A perfectly rigid car transmits all force to passengers. Safety comes from controlled deformation, not rigidity.

Question 1 of 5

A 1000 kg car traveling at 10 m/s crashes and stops. If the collision lasts 0.1 s, what is the average force on the car?

Question 2 of 5

A safety engineer is redesigning a car's front end. Which single change to the collision itself would reduce the impact force on a passenger by half?

Question 3 of 5

Why are car crumple zones made of deformable metal rather than ultra-rigid steel?

Question 4 of 5

In an elastic collision, kinetic energy is conserved. Why might this be worse for passenger safety compared to an inelastic collision?

Question 5 of 5

A 0.5 kg ball hits a wall at 6 m/s and bounces back at 4 m/s. What is the magnitude of the impulse on the ball?