Life-saving shock absorbers

Problem

Life-saving shock absorbers. Explain why, in a head-on collision, shock absorbers and controlled bodywork deformation save lives. Argue in terms of impulse, collision duration and average force on the passenger.

Solution

In the collision, the change in the passenger’s momentum is fixed: it depends only on the mass and the initial velocity, which must in any case go to zero. This change is the required impulse: $J = \Delta p = m\,\Delta v \quad (\text{fixed, independent of how braking happens})$ Impulse is also average force times duration: $J = F_\text{avg}\,\Delta t \quad\Rightarrow\quad F_\text{avg} = \frac{J}{\Delta t} = \frac{\Delta p}{\Delta t}$ Since $\Delta p$ is fixed, the average force on the passenger is inversely proportional to the duration of the collision: stretching $\Delta t$ reduces $F_\text{avg}$.

Without shock absorbers or deformation, the stop is almost instantaneous, $\Delta t \sim 2\;\text{ms}$, and the force is enormous. With crumpling bodywork and shock absorbers, the stopping time rises to $\Delta t \sim 50\;\text{ms}$: the same impulse is spread over a time about $25$ times longer, so the peak force drops by about $25$ times, within the body’s survival threshold. Result: $\ev{\text{same } J, \text{ but } \Delta t \times 25 \Rightarrow F_\text{avg} \div 25}$

Links

Topics: Momentum and collisions Concepts: Impulse · Momentum Skills: Symbolic setup