The car stops either way. So why does the crash hit you so much less hard with a crumple zone?
The answer is one single idea: momentum. Try it out.
What is momentum?
A truck rolling slowly has more "oomph" than a fast tennis ball. That oomph depends on two things: how heavy something is and how fast it moves.
Physicists call it momentum and write it like this: p = m·v, mass times velocity. The unit is kg·m/s.
Direction counts. A ball flying left has negative momentum. Two equal balls flying towards each other add up to zero.
Momentum gets handed on.
Five steel balls hang in a row. The toy is called Newton's cradle. Guess first, then try it.
What do you think: you pull 2 balls out. How many fly off on the other side?
Per-ball numbers
Balls of 0.10 kg on 0.20 m strings; small-angle period T ≈ 2π√(L/g) = 0.90 s. Released from 30°, energy conservation gives an impact speed of 0.725 m/s. Last hit, momentum: – → – kg·m/s, kinetic energy: – → – mJ.
- What you saw: Two balls in, two balls out. At the same speed.
- Why: The momentum runs through the row. Each ball hands it to the next, and two equal balls simply swap speeds when they hit.
- Honestly: A real cradle slowly calms down. Every hit turns a little energy into sound and warmth.
Why not 4 balls, just slower?
Four balls at half speed would carry exactly as much momentum as two at full speed: 4 · m · v/2 = 2 · m · v. So momentum alone does not forbid it. But motion has a second quantity: energy. It grows with the square of the speed. Four balls at half speed would have only half the energy: 4 · ½m(v/2)² = ½mv², but 2 · ½mv² = mv² came in.
Hard steel balls lose almost no energy, so any outcome must keep both momentum and energy. That rules out two at half speed. But the two laws alone do not fix the outcome for five balls: other splits keep both as well. What selects one ball out (and two for two) is how the push travels. Hard balls that just touch pass it on almost like a chain of separate two-ball collisions, and in an elastic collision between two equal balls they simply swap velocities. The demo above is built on exactly that model.
Two caveats. A real cradle loses a little on every hit, to sound and heat, and slowly dies down. And the clean one-in, one-out relies on the balls being hard and just touching. Switch on the losses in the demo above, and after a few swings the other balls start to drift along.
How hard it hits you.
Catch a raw egg. You pull your hand back as you catch it, so the egg slows down over a longer time, and it survives. Jump off a wall: you bend your knees so the stop takes longer. A bike helmet has foam for the same reason. Always the same idea: you have to get rid of the momentum anyway. More time for it, less force.
That is all. The rest is arithmetic. In a 50 km/h crash you (75 kg) must get rid of 1042 kg·m/s of momentum. That is fixed, whatever the car is built of. The crumple zone only gives you more time for it.
Your crash test
Drag the speed, then crash- Impact
- Front crumples
- Belt holds
- Airbag catches head
- Dashboard
- all still
What do you think: without the crumple zone, does the force double, triple, or more?
The timings in detail
50 km/h is 13.9 m/s. We treat each stop as a steady (constant) deceleration, an idealisation: real crash pulses have peaks. Then the stopping time is Δt = 2·Δd / v.
The front crushes 0.6 m, so the car stops in 2 × 0.6 / 13.9 ≈ 86 ms. Belt and airbag let the occupant move about 0.3 m further forward relative to the car while they slow down, so the occupant stops over about 0.9 m, in 2 × 0.9 / 13.9 ≈ 0.13 s. The belt tightens and the airbag fills within the first few hundredths of a second.
All numbers
| With crumple zone | No crumple zone |
|---|
Constant-deceleration model: every force is an average. Car mass 1200 kg (assumed), Earth 5.97 × 10²⁴ kg. Horizontal lengths in the drawing are to scale.
What you now know.
- Momentum gets handed on, never destroyed.
- How fast it gets handed on, that is the force.
- More time or more distance, less force: crumple zone, belt, airbag, helmet, bent knees.
Where does the car's momentum go?
Into the wall, and through its foundations into the Earth. Car plus Earth is a closed system, so its total momentum does not change. A 1200 kg car at 13.9 m/s carries about 16 700 kg·m/s. Shared with the Earth's 6 × 10²⁴ kg, that is a speed of about 3 × 10⁻²¹ m/s: real, and far too small to ever measure. And it is not new: when the car sped up, its tyres pushed the Earth the other way by exactly this much. The crash simply returns it.
That ties the two halves together. Momentum is never lost, it is only handed on. How fast it is handed on is the force. The cradle hands it on in a flash between hard balls. A crumple zone and a belt hand it on slowly, and that is the whole trick.
The exact figures
The change of momentum is exactly what Newton's second law is about. For constant mass, F = m·a = m·Δv/Δt is the same statement. The product F·Δt is called the impulse, and it always equals Δp.
Δp is fixed, so only Δt decides the force. Stretch the stop from 7 ms to 0.13 s and the force drops about 18 times. The crumple zone is a length, not a time, so it enters through the energy: the force times the stopping distance must soak up the kinetic energy, F·Δd = ½·m·v². For a steady stop, Δt = 2·Δd/v. More metres, more milliseconds, less force.
Same 75 kg, same 50 km/h, same Δp of 1042 kg·m/s in all three. Average force; "g" is the deceleration in multiples of gravity, which is also the force in multiples of your weight.
Assumptions. Rigid wall, head-on. Occupant 75 kg, 50 km/h = 13.9 m/s. Every stop is a constant deceleration, so the forces are averages: real crash pulses rise and fall, and their peaks are higher. Crush 0.6 m with a crumple zone, 0.05 m without. Restraint ride-down (how far the occupant moves forward relative to the car while being held): belt 0.25 m, belt + airbag 0.30 m. Without a belt the occupant is not tied to the car: they fly on at full speed until they hit the dashboard, and stop there over about 0.05 m. With the dashboard 0.65 m ahead, more than the 0.6 m crush, the car has already stopped when they arrive, so here the crumple zone does not help them. The restraint is assumed to hold from the first instant; real belts have slack, so the real peak is later and higher.
Formulas. Δp = m·v. Δt = 2·Δd/v. F = Δp/Δt = m·v²/(2·Δd). Deceleration in g: v²/(2·Δd·9.81). Check: 8.04 kN × 0.9 m = 7.23 kJ = ½ × 75 kg × (13.9 m/s)².
The airbag helps twice. It lengthens the head's stopping distance and time, and it spreads the force over a much larger area, so the pressure on any one spot is lower. F = Δp/Δt only captures the first.
History. Newton stated his second law in terms of the change of "motion", his word for momentum, in the Principia (1687). The cradle figures are illustrative: steel balls of 0.10 kg on 0.20 m strings.
Next step for this series: every case becomes a sourced, time-stamped fact you can question in the graph.
See how the graph works →