Crash test with airbag

Problem

During a safety test, a car of $1\,200\;\mathrm{kg}$ hits a wall at $5{,}0\;\mathrm{m/s}$ and stops in $0{,}50\;\mathrm{s}$. The airbag, however, reduces the stopping time for the driver ($75\;\mathrm{kg}$) to about $0{,}30\;\mathrm{s}$, against $0{,}10\;\mathrm{s}$ without the airbag. Estimate the average force acting on the driver with the airbag and evaluate the benefit.

Solution

I focus the analysis on the driver, of mass $m = 75\;\mathrm{kg}$, who goes from $v = 5{,}0\;\mathrm{m/s}$ to rest. Their change in momentum (in magnitude) is: $\abs{\Delta p} = m\,v = 75\;\mathrm{kg}\cdot 5{,}0\;\mathrm{m/s} = 375\;\mathrm{kg\,m/s}$ By the impulse-momentum theorem the average force is $F = \abs{\Delta p}/\Delta t$. With the airbag ($\Delta t = 0{,}30\;\mathrm{s}$): $F_{\text{airbag}} = \frac{375\;\mathrm{kg\,m/s}}{0{,}30\;\mathrm{s}} = 1{,}25\cdot 10^3\;\mathrm{N} \approx 1{,}2\cdot 10^3\;\mathrm{N}$ Without the airbag ($\Delta t = 0{,}10\;\mathrm{s}$): $F_{\text{senza}} = \frac{375\;\mathrm{kg\,m/s}}{0{,}10\;\mathrm{s}} = 3{,}75\cdot 10^3\;\mathrm{N}$ The gain is the ratio between the two forces: $\frac{F_{\text{senza}}}{F_{\text{airbag}}} = \frac{0{,}30}{0{,}10} = 3$ The airbag, by tripling the stopping time, reduces the average force on the driver to a third. $\ev{F \approx 1{,}2\cdot 10^3\;\mathrm{N}\ \ (\text{about 3 times less than without the airbag})}$

Links

Topics: Quantità di moto e urti Concepts: Impulso · Quantità di moto Skills: Impostazione simbolica