Question

2.23 •• bio automobile airbags. the human body can survive an acceleration - trauma incident (sudden stop) if the magnitude of the acceleration is less than 250 m/s². if you are in an automobile accident with an initial speed of 105 km/h (65 mi/h) and are stopped by an air - bag that inflates from the dashboard, over what minimum distance must the airbag stop you for you to survive the crash?

Explanation:

Step1: Convert initial speed to m/s

The initial speed $v_0 = 105\ km/h$. To convert it to m/s, use the conversion factor: $1\ km = 1000\ m$ and $1\ h=3600\ s$. So $v_0 = 105\times\frac{1000}{3600}\ m/s=\frac{1050}{36}\ m/s\approx 29.17\ m/s$. The final - speed $v = 0\ m/s$ and the maximum acceleration $a=- 250\ m/s^{2}$ (negative because it is decelerating).

Step2: Use the kinematic equation

The kinematic equation $v^{2}=v_0^{2}+2ax$ is used, where $v$ is the final velocity, $v_0$ is the initial velocity, $a$ is the acceleration, and $x$ is the displacement. Rearranging the equation for $x$ gives $x=\frac{v^{2}-v_0^{2}}{2a}$.
Substitute $v = 0$, $v_0\approx29.17\ m/s$, and $a=-250\ m/s^{2}$ into the equation:
\[

$$\begin{align*} x&=\frac{0-(29.17)^{2}}{2\times(-250)}\\ &=\frac{- 850.89}{-500}\\ & = 1.70178\ m\approx1.70\ m \end{align*}$$

\]

Answer:

$1.70\ m$