Why airbags save lives

Problem

Why do airbags save lives? Explain in terms of impulse and momentum. Why is a wall less dangerous than another vehicle travelling in the opposite direction, at the same speed?

Solution

Airbag. In a collision the body must lose all its momentum: the change $\Delta p$ is fixed by the mass and the initial velocity. The impulse-momentum theorem links this change to the average force and the stopping time: $\bar F \, \Delta t = \Delta p \;\Longrightarrow\; \bar F = \frac{\Delta p}{\Delta t}$ With $\Delta p$ fixed, lengthening the braking time $\Delta t$ reduces the average force $\bar F$ on the body. The airbag (like the seatbelt) does exactly this: it spreads the stop over a longer time and a larger area, reducing the peak force that would otherwise be discharged onto the skull against the steering wheel.

Wall versus vehicle. At the same speed $v$:

• against a wall the car goes from $v$ to $0$: $|\Delta v| = v$;
• against an identical car arriving at $-v$, in the collision (perfectly inelastic) both stop at the centre of mass, so each one goes from $v$ to $0$… but if we think of the head-on collision as “hitting a wall that comes towards you”, the relative impact speed is $2v$: the change in momentum involved is twice that against a stationary wall.

So $\Delta p$ (and with it $\bar F$) is greater against a vehicle travelling in the opposite direction: it is more dangerous than the wall.

Final result: $\ev{\bar F = \Delta p / \Delta t:\ \text{the airbag increases}\ \Delta t;\ \text{head-on collision} \Rightarrow \Delta v\ \text{doubled} \Rightarrow \bar F\ \text{greater}}$

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Topics: Quantità di moto e urti Concepts: Impulso · Quantità di moto