Grade by Grade
C. Energy and Momentum · Momentum and Collisions

Momentum and Safety: Vehicles, Helmets and Sports Equipment

C1.1 C1.2 C2.5 A2.1About 70 minutes
Learning goals
  • I can analyse a safety technology using impulse, work and kinetic energy, pressure, and rebound.
  • I can explain why crumple zones, vented airbags and crushable foams are designed to be inelastic.
  • I can explain how sports equipment increases the impulse or energy given to a ball or puck.
  • I can propose a specific, testable improvement to protective or sports equipment and describe its trade-offs.

Words to know

crumple zone
A part of a vehicle designed to crush in a controlled way, lengthening the stopping distance and time.
passenger cell
The rigid central structure of a vehicle that keeps a survival space around the occupants.
peak force
The largest force reached during an impact; often more important for injury than the average force.
load limiter
A seat-belt device that lets the belt spool out slightly once the force reaches a set level, flattening the force-time curve.
pressure
Force per unit area, P = F ÷ A, measured in pascals (Pa).

Designing for the worst moment

This lesson is about how engineers protect people. Collisions can be a hard topic for anyone who has been affected by one; the focus here is on the physics of keeping people safe.

Remember from Grade 11: in the SPH3U Forces unit you saw that seat belts, airbags and crumple zones make a stop last longer, so the force is smaller (F = mΔv ÷ Δt). In this course you have much sharper tools: impulse and force–time graphs, work and kinetic energy (from the sibling unit Work, Energy and Simple Harmonic Motion), pressure, and the difference between elastic and inelastic collisions. Your curriculum asks you to use them to analyse a technology, such as protective equipment, and to propose ways to improve it. This lesson does that for vehicle safety and sports equipment.

Four tools for protecting a body

When a person must be stopped, their change in momentum Δp and their kinetic energy are fixed by their mass and speed. A safety system cannot change these. It can only change how they are removed:

  1. Spread it over time (impulse). F_avg = Δp ÷ Δt. A longer stop means a smaller average force.
  2. Spread it over distance (work). F_avg × d = ΔEk. A longer stopping distance means a smaller average force. Crumple zones and airbags add stopping distance.
  3. Spread it over area (pressure). Pressure = F ÷ A. A seat belt across the hips and chest, a wide airbag or a hard helmet shell spreads the force over a large area of the body's strongest parts.
  4. Avoid bouncing (inelastic is good here). A bounce adds extra Δp in the opposite direction, so it increases the impulse and force. Airbags have vents that let gas escape as the occupant presses in, and helmet foams crush instead of springing back. Here, an inelastic collision is exactly what engineers want.

A good design also keeps the force-time graph low and flat, avoiding a sharp peak. The peak force, not just the average, often decides whether bones break.

204060801005101520time (ms)force on occupant (kN)A: no restraint, 20 kN peakB: belt + airbag, 7.5 kN peakC: ideal flat top, 4.3 kNAll three areas = 300 N·s (the same change in momentum)
Three ways of removing the same momentum (300 N·s, a 60 kg person stopping from 5 m/s, drawn as an example). The areas are equal, but spreading the stop over a longer time with a flat top keeps the peak force lowest. Modern seat belts use load limiters to flatten the peak.
Worked example

Example 1: Crumple zone versus dashboard

A 60 kg occupant is in a car travelling at 50 km/h that hits a solid barrier. Find the average force on the occupant if (a) a seat belt and crumple zone stop the occupant over 0.50 m, (b) an unbelted occupant hits the dashboard and stops over 0.050 m. Also estimate each stopping time.

  1. v = 50 ÷ 3.6 ≈ 13.9 m/s. Ek = ½(60)(13.9)² ≈ 5.79 × 10³ J.
  2. (a) F_avg = Ek ÷ d = 5790 ÷ 0.50 ≈ 1.16 × 10⁴ N. Time (uniform deceleration, average speed 6.94 m/s): 0.50 ÷ 6.94 ≈ 0.072 s.
  3. (b) F_avg = 5790 ÷ 0.050 ≈ 1.16 × 10⁵ N. Time: 0.050 ÷ 6.94 ≈ 0.0072 s.
  4. Check with impulse: Δp = (60)(13.9) ≈ 833 N·s; 833 ÷ 0.072 ≈ 1.16 × 10⁴ N ✓.

(a) About 1.2 × 10⁴ N (about 20 times the occupant's weight). (b) About 1.2 × 10⁵ N, ten times larger, usually far beyond what the body can tolerate.

front crumplezonerigid passenger cell(survival space)rear crumplezonecrushes on purpose:longer stop, smaller forceprotects inrear-end collisionsSoft ends absorb kinetic energy by deforming; the strong middle keeps its shape.
A modern car body has soft ends and a hard middle. The crumple zones crush in a controlled way to lengthen the stop (an intentionally inelastic collision), while the rigid passenger cell protects the space around the occupants.
Worked example

Example 2: Why airbags must not bounce

An occupant's head and upper body (mass 5.0 kg in this model) moving at 10 m/s are stopped by an airbag in 0.050 s. Compare the average force if (a) the airbag vents and the body stops, (b) a non-venting airbag makes the body rebound at 4.0 m/s in the same time.

  1. Forward positive. (a) Δp = 5.0(0 − 10) = −50 N·s. F = 50 ÷ 0.050 = 1000 N.
  2. (b) Δp = 5.0(−4.0 − 10) = −70 N·s. F = 70 ÷ 0.050 = 1400 N.

The bounce raises the force by 40% (1400 N instead of 1000 N). That is why airbags vent and helmet liners crush: making the collision inelastic protects the person.

Worked example

Example 3: Spreading the force

A force of 4000 N acts on a skull. Compare the pressure if it acts on an area of 4.0 cm² (a small hard object) and if a helmet shell spreads it over 100 cm².

  1. 4.0 cm² = 4.0 × 10⁻⁴ m², so P = 4000 ÷ 4.0 × 10⁻⁴ = 1.0 × 10⁷ Pa.
  2. 100 cm² = 1.0 × 10⁻² m², so P = 4000 ÷ 1.0 × 10⁻² = 4.0 × 10⁵ Pa.

The shell cuts the pressure 25 times (from 1.0 × 10⁷ Pa to 4.0 × 10⁵ Pa), which helps prevent the skull from fracturing. The foam liner behind the shell then lengthens the stop to reduce the force itself.

Helmets and pads

  • Bicycle helmets usually have a liner of expanded polystyrene (EPS) foam. It crushes permanently in a hard impact, absorbing energy (very inelastic), but it cannot do this twice. That is why manufacturers say to replace a helmet after any crash, even if it looks fine.
  • Hockey helmets use foams designed to recover their shape for repeated impacts, and must be certified to a safety standard (in Canada, CSA-certified helmets are required in organized minor hockey).
  • Goalie pads, shin guards and gym mats combine a hard or tough outer layer (spreads area) with soft padding (adds time and distance).
  • Running shoes lengthen the landing slightly, lowering the peak force on joints.

Sports equipment that does the opposite

Some equipment is designed to increase the momentum given to a ball:

  • Golf drivers have thin faces that flex and spring back, raising the coefficient of restitution of the club-ball collision. Golf's rule-makers limit how "springy" a driver face may be, because it adds distance.
  • Hockey sticks flex during a slapshot, so the blade stays on the puck longer (bigger impulse).
  • Tennis racket strings and trampolines store energy elastically and return most of it, rather than turning it into thermal energy.
Worked example

Example 4: A springier club face

For a club head of mass m₁ moving at v₁ hitting a ball of mass m₂ at rest, momentum and the coefficient of restitution e give the ball speed v₂′ = (1 + e)m₁v₁ ÷ (m₁ + m₂). Find the ball speed for a 0.20 kg club head at 45 m/s and a 0.046 kg ball when e = 0.70 and when e = 0.83.

  1. m₁v₁ ÷ (m₁ + m₂) = (0.20)(45) ÷ 0.246 ≈ 36.6 m/s.
  2. e = 0.70: v₂′ = 1.70 × 36.6 ≈ 62.2 m/s.
  3. e = 0.83: v₂′ = 1.83 × 36.6 ≈ 67.0 m/s.

About 62 m/s and 67 m/s: the springier face adds almost 5 m/s, which can mean many metres of extra distance. (The formula is derived in the Challenge.)

Proposing improvements

To propose an improvement to a safety or sports technology:

  1. State its job in physics terms (for example, "stop a 5 kg head from 6 m/s with a peak force under a set limit").
  2. Identify the limits of the current design (it bottoms out at high speed; it bounces; it is too heavy; it works only once).
  3. Suggest a change using the four tools: more time or distance, more area, less rebound, a flatter force-time curve.
  4. Consider trade-offs: mass, cost, comfort, durability, and whether people will actually use it.
  5. Say how you would test it (drop tests, force sensors, crash tests with instrumented dummies, comparing peak forces).

Careers that use this physics include automotive safety engineers, biomedical engineers, kinesiologists and biomechanists, collision reconstructionists, and sports equipment designers. Most begin with a university degree in engineering, physics or kinesiology, for which SPH4U and calculus are usual prerequisites.

Just for fun

Helmet testers drop helmets onto steel anvils with a metal "headform" inside, packed with sensors. It may be the only head in the world that has been hit thousands of times and still never complains about helmet hair.

Hands-on activity

Egg drop: analyse, test and improve

You need: raw eggs (3 or 4), and household materials such as paper, cardboard, drinking straws, tape, cotton balls, a plastic bag, string and rubber bands; a tape measure; zip-top bags. Grown-up note: drop from a safe spot (for example, standing on the floor and dropping from 2.0 m, or from a deck with a railing, never from a window or roof); wash hands after handling raw eggs.

  1. Design: build a protective package for one raw egg (sealed in a zip-top bag) using no more than 10 straws, 1 m of tape and 2 sheets of paper. Your package must be dropped, not thrown.
  2. Analyse before you test: write down which of the four tools (time, distance, area, no bounce) your design uses, and where.
  3. Test: drop it from 2.0 m onto a hard floor. Record what happened. Film it in slow motion if you can, and count the frames from first contact to stopping.
  4. Improve: change one thing and test again. Keep going until the egg survives, then try a greater height.
  5. Report: explain your final design with impulse and energy, and list one trade-off (for example, a bigger package is safer but uses more material).

What to look for: successful designs usually crush or deform (straw cages, crumpled paper), slow the egg over several centimetres, hold it so the force is spread around the shell, and do not bounce. A parachute reduces the impact speed itself, which reduces Δp.

Your turn

1

An 80 kg occupant travelling at 20 m/s is stopped over 0.60 m by a belt and crumple zone. Find the average force and the stopping time.

2

Repeat the previous question for a stopping distance of 0.050 m.

3

A force of 2000 N acts over 2.0 cm². Find the pressure, and the pressure if a pad spreads it over 50 cm².

4

A 5.0 kg head moving at 8.0 m/s is stopped in 0.040 s. Find the average force if it stops, and if it rebounds at 3.0 m/s in the same time.

5

Why should a bicycle helmet be replaced after a crash, even if it looks undamaged?

6

Explain, using impulse, how a flexing hockey stick helps produce a faster shot.

7

Name two careers that use the physics of momentum and impulse to protect people.

More practice: warm-up

1

Can a seat belt change a person's change in momentum in a crash?

2

Why do crumple zones make the car's collision more inelastic on purpose?

3

What does a hard helmet shell do?

4

Why do airbags have vents?

5

Give an example of sports equipment designed to increase the impulse on a ball or puck.

More practice: on your own

1

A 70 kg runner lands with a vertical speed of 3.0 m/s. Find the average force of the ground if the landing lasts 0.010 s on a hard surface, and 0.030 s with cushioned shoes. Include the runner's weight.

2

Using v₂′ = (1 + e)m₁v₁ ÷ (m₁ + m₂), explain why a clubhead can never launch a ball faster than (1 + e) times the clubhead's speed, and find that limit for e = 0.83.

3

Propose one improvement to a youth hockey helmet and describe how you would test it.

4

A crumple zone is made much softer. Explain why this might increase the peak force on occupants in a high-speed crash.

5

Why do infant car seats face backward?

Challenge

1

Derive v₂′ = (1 + e)m₁v₁ ÷ (m₁ + m₂) for a head-on collision with object 2 at rest, using momentum and e = (v₂′ − v₁′) ÷ v₁.

2

Two safety systems each provide an impulse of 80 N·s over 0.040 s. One gives a triangular force-time pulse, the other a flat-topped (rectangular) pulse. Find the peak force of each.

3

A seat-belt load limiter keeps the belt force at no more than 6000 N. A 75 kg occupant must be stopped from 15 m/s by the belt alone. Find the shortest possible stopping time and the distance the occupant moves relative to the ground during it (constant force).

Check your understanding

Quiz

1.

A crumple zone reduces the force on the occupants mainly by:

2.

If an airbag made the occupant bounce back instead of stopping, the average force on the occupant (for the same time) would be:

3.

The hard outer shell of a helmet mainly:

4.

Why should a bicycle helmet with an EPS foam liner be replaced after a crash?

5.

A golf driver face with a higher coefficient of restitution will:

6.

Why is the passenger cell of a car designed to be rigid while the front is designed to crush?

For parents and teachers

Big idea (C1.1, C1.2, C2.5): a safety system cannot change a person's change in momentum or kinetic energy in a crash, but it can spread them over more time, more distance and more area, and avoid rebound. Sports equipment often does the reverse, to increase the impulse on a ball. Your teen practised analysing a technology and proposing improvements, with trade-offs and a test plan.

A note for parents: collisions can be a sensitive topic. If your family has been affected by a crash or a sports injury, it is fine to focus on the egg-drop design and the physics.

How to help: Do the egg drop together and ask your teen to explain each design choice with the four tools (time, distance, area, no bounce).

Common difficulties:

  • Saying safety devices "reduce momentum". They reduce the force, by stretching out the change in momentum.
  • Forgetting rebound. A bouncy collision is worse for safety.
  • Vague proposals. A good proposal names the physics goal, a specific change, a trade-off, and a test.
  • Units of area: 1 cm² = 1 × 10⁻⁴ m².