Car braking and the seatbelt

Problem

When a car brakes sharply, people are pushed forward. Explain in terms of Newton’s laws why this happens. Is it really a “forward force” that pushes them? Why does the seatbelt save lives?

Solution

There is no forward force at all. The sensation of being “pushed forward” is misleading: no real force acts forward on the passenger. The phenomenon is governed by the first law (principle of inertia). Before braking, car and passenger travel together at speed $v$. When the driver brakes, the brakes apply a force (brake-wheel friction, then wheel-road friction) that decelerates the car. But on the passenger’s body, until a constraint intervenes, no horizontal force acts to slow them down.

By inertia, therefore, the passenger tends to continue in a straight line at the initial speed $v$, while the cabin slows down beneath them. In the car’s (non-inertial) reference frame it looks as if the passenger is being flung forward; in the ground’s inertial frame, however, it is simply the car that “slides backward” relative to them. The “forward force” is an apparent force, not a real one.

The role of the seatbelt. For the passenger to slow down together with the car, a real external force directed backward is needed. It is precisely the seatbelt (together with the seat back and, in a crash, the airbag) that supplies it: it applies to the chest and pelvis the decelerating force that inertia alone does not produce. Without a seatbelt, the body continues undisturbed until it meets a rigid obstacle — steering wheel, dashboard, windscreen — that stops it in a very short time, with enormous forces.

Why it saves lives. The seatbelt acts on two fronts linked to the impulse relation, force $\times$ time $=$ impulse, $\bar F\,\Delta t = \Delta p$. For a given change in momentum $\Delta p$ to be absorbed:

• it lengthens the stopping time $\Delta t$ (the body decelerates together with the car, not against an instantaneous obstacle), reducing the average force $\bar F$;
• it spreads the force over large, sturdy areas of the body (pelvis and collarbone) instead of concentrating it on the head or chest against the dashboard.

$\ev{\text{Inertia (1st law): the body continues; the seatbelt is the external force that decelerates it safely.}}$

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Topics: Dynamics Concepts: Newton’s first law · Newton’s second law