Why Do Airbags Save Lives?

A bus and a motorbike are coming towards you at the same speed. Which would you rather try to stop? Not the bus: it has far more mass. Now a cricket ball, tossed gently to you, or the same ball from a fast bowler? The fast one is much harder to stop. How hard a moving thing is to stop depends on both its mass and its velocity.

Physicists put the two together. The momentum of a body is its mass times its velocity:

$$p = mv$$

where p is the momentum, m the mass in kilograms and v the velocity in meters per second. So momentum is measured in kilogram meters per second, kg m/s.

Moving thing	Mass	Velocity	Momentum
a rifle bullet	0.010 kg	800 m/s	8 kg m/s
a cricket ball from a fast bowler	0.16 kg	40 m/s (144 km/h)	6.4 kg m/s
you, walking	60 kg	1.5 m/s	90 kg m/s
a motorbike and its rider	200 kg	14 m/s (50 km/h)	2,800 kg m/s
a loaded bus	12,000 kg	11 m/s (40 km/h)	132,000 kg m/s

A bullet is tiny but very fast; a bus is slow but enormous. Momentum puts them on one scale.

You met velocity in Grade 9: it is speed in a given direction Grade 9: Speed and velocity. Momentum has a direction too, the direction of the velocity. On a straight road, choose one direction as positive (say, to the right, or east); then anything moving the other way has a negative velocity, and a negative momentum. Two trolleys of 1 kg, one moving right at 2 m/s and one moving left at 2 m/s, have momenta of +2 and −2 kg m/s: the same size, opposite directions.

You learned Newton’s second law in Grade 9 as F = ma Grade 9: Newton’s second law of motion. Acceleration is the change in velocity divided by the time taken, $a = \dfrac{v - u}{t}$. Put that in:

$$F = m\,\frac{v - u}{t} = \frac{mv - mu}{t}$$

mu is the momentum at the start and mv the momentum at the end. So

$$F = \frac{\text{change in momentum}}{\text{time taken}} \qquad \text{or} \qquad F\,t = mv - mu$$

The force on a body equals the rate at which its momentum changes. This is actually how Newton himself wrote his second law, in 1687: “the alteration of motion is ever proportional to the motive force impressed” (in the English translation of 1729), where by “motion” he meant mass times velocity, what we call momentum.

The force multiplied by the time it acts, Ft, is called the impulse. It is measured in newton seconds, N s, which is the same as kg m/s. The rule Ft = mv − mu says: the impulse on a body equals its change in momentum.

Read it backwards and you have the secret of this whole topic. To stop something, you must take away all its momentum. That change is fixed: it is the momentum it arrived with. But you can choose how long the stop takes. A short stop needs a big force; a long stop needs only a small one:

$$F = \frac{\text{change in momentum}}{t}$$

In a real stop, the force isn’t steady: it rises and falls. Then the impulse is the area under the graph of force against time, and the force worked out from F = Δp ÷ t is the average force. The largest force, the peak, is what breaks eggs and bones, so it matters even more.

Now think about two things that hit each other: a moving trolley running into one at rest, a bat meeting a ball, two buses. Newton’s third law Grade 9: Newton’s third law of motion says that while A pushes B, B pushes A back, with a force equal in size and opposite in direction. And since each force exists only while A and B touch, the two forces act for exactly the same time. They start when A and B first touch and stop when they part.

So the impulse on B (force × time) is equal and opposite to the impulse on A. That means:

$$\text{change in B’s momentum} = -\,\text{change in A’s momentum}$$

Whatever momentum A loses, B gains, to the last bit. Add them up, and the total momentum of A and B together hasn’t changed at all. This is the law of conservation of momentum:

When no outside force acts on a group of bodies, their total momentum stays the same, whatever they do to each other.

The group of bodies we are looking at is called a system, and a force from something outside the system is an external force. The forces A and B exert on each other are internal: they only move momentum from one to the other. For two bodies in a straight line, with velocities u₁ and u₂ before and v₁ and v₂ after:

$$m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2$$

Before you try it with any masses and speeds you like, one more example, with things moving in opposite directions. Then the signs matter.

Momentum is conserved even when things start at rest and fly apart. Their total momentum was zero, so it must stay zero: the two pieces go off with equal and opposite momentum. That is the recoil of a gun, the kick of a balloon rocket from Grade 9, and the reason a rocket can move forward by throwing gas backwards, with nothing to push against.

A law in science is only as good as the evidence for it. Here is how a class can test conservation of momentum, and the kind of results it gets.

Two trolleys run on a level track. Trolley 1 is pushed towards trolley 2, and a motion sensor (or a phone video, measured frame by frame) gives each trolley’s velocity just before and just after they collide and stick together. Here are five runs, typical of what a class gets (made-up numbers, not one real class’s measurements), with velocities to the right as positive:

Run	m₁ (kg)	u₁ (m/s)	m₂ (kg)	u₂ (m/s)	v after, stuck together (m/s)
1	0.50	0.80	0.50	0	0.39
2	1.00	0.60	0.50	0	0.39
3	0.50	0.90	1.00	0	0.29
4	1.00	0.50	1.00	−0.30	0.09
5	1.50	0.40	0.50	−0.60	0.14

Back to the question we started with. In a crash, your body arrives with a certain momentum, and it must all be taken away: the crash decides the change in momentum, not you. The only thing that can be changed is how long the stop takes, and so, by F = Δp ÷ t, how big the force is. Every safety device in this section works by making the stop longer. (The other way is to have less momentum to lose in the first place: drive slower, or slow down before the hit, as the brakes do, or as a parachute does for a falling skydiver, Grade 10.)

A longer stop also means a longer stopping distance: the body moves while it slows down. A helmet’s foam gives the head a few centimeters to stop in instead of a few millimeters. A car’s front gives the passengers most of a meter.

• Crumple zone. The front of a car is designed to crush, folding up like an accordion. While it crushes, the strong passenger section keeps moving forward, slowing steadily, instead of stopping dead. A car that didn’t crumple would look better after a crash, and the people inside would be worse off. The idea was patented by Béla Barényi, an engineer at Mercedes-Benz, in 1952, and first built into a car in 1959.
• Safety cell. The passenger section itself is made rigid, so that it doesn’t crush onto the people inside.
• Seatbelt. It holds you to the car, so that you slow down with it, over the whole crumpling time, instead of flying on and stopping in a hundredth of a second against the dashboard. It is made to stretch a little, adding time. Wearing one reduces the risk of death in a crash by up to half, according to the WHO.
• Airbag. It gives the head, which is heavy and moves on a bendy neck, something soft to sink into, so it stops over many centimeters instead of a few millimeters against the steering wheel.
• Helmet. Its foam liner crushes, so the head stops over a centimeter or two instead of a millimeter or so on the road. Its hard shell spreads the force over a wide area of the foam and the skull.

The same idea is everywhere, once you look for it. Long jumpers land in a pit of soft sand; pole vaulters and high jumpers on thick foam mats. Eggs and phones are packed in molded thermocol or cardboard. A goalkeeper’s gloves are padded; a cricket batter’s helmet is lined with foam. Gymnasium floors are sprung. A climbing rope stretches when it catches a fall, and it is made to.

An airbag has to do something remarkable: fill up between your head and the steering wheel after the crash has started, but before your head gets there. The whole crash is over in about a tenth of a second.

1. A sensor in the car measures how fast it is slowing down. Braking hard slows a car at about 1 g; a crash slows it at 20 g or more. Within about 15 to 30 thousandths of a second (milliseconds, ms) of the crash starting, the car’s computer decides that this is a serious crash.
2. It sends an electric current to an igniter, which sets off a small charge of chemicals in the inflator. They react in a flash and make a large volume of gas: in the first airbags, sodium azide breaking up into sodium and nitrogen gas, 2NaN₃ → 2Na + 3N₂, with other chemicals added to make the sodium harmless. Most airbags today use other chemicals, such as guanidine nitrate, because sodium azide is poisonous.
3. The gas fills the folded nylon bag in about 30 ms, bursting it out through the cover of the steering wheel.
4. Your head, still moving forward, meets the bag. The bag has vent holes, and your head pushes the gas out through them, so the bag gives way steadily under you instead of bouncing you back like a balloon. Your head sinks in and slows down over many centimeters.
5. A moment later the bag is empty and limp, so you can see and get out.

How do engineers design a helmet, or an airbag, or packing for a phone? Not by guessing once. They follow a cycle that every kind of engineer uses:

1. Define the problem. What must the device do? These are its criteria: a helmet must keep the force on the head below a safe limit. What limits the design? These are its constraints: it must be light and small enough to wear, cheap enough to buy, and pass the official test.
2. Imagine and plan. Think of several ways to do it, and choose one, using what you know (here: a longer stop means a smaller force).
3. Build and test a model.
4. Evaluate: did it meet the criteria? Where did it fail?
5. Improve, changing one thing at a time, and test again.

A bicycle helmet sold in Europe must pass a test like this. A metal test head of about 5 kg, wearing the helmet, is dropped about 1.5 m onto a steel block, hitting it at 5.42 m/s. The helmet fails if the head is ever slowed down at more than 250 g, that is, if the force on it is more than 250 times its own weight.

The liner of a helmet is expanded polystyrene (EPS), the same white bubbly foam as thermocol packing. When it is pressed hard enough, its tiny bubbles collapse, and it crushes at a steady force, using up the head’s energy as it goes. It doesn’t spring back. That’s exactly what you want for a smooth stop, but it means a helmet that has been in a crash has done its job and must be replaced, even if it looks fine.

• Why is spider silk so strong? A spider’s web stops a flying insect without snapping. How the shapes of molecules make a material strong, or stretchy, or both.
• Where does the energy go when a ball bounces? In a crash, momentum is kept but kinetic energy disappears. Follow the energy into heat and sound, and watch the total stay the same.
• Why is gravity so weak? The forces that push trolleys apart in a collision are electric forces between atoms. How strong they are, compared with gravity.