Question 2.11: Force exerted by air bag on driver during collision A 60-kg ......
Force exerted by air bag on driver during collision
A 60-kg crash test dummy moving at 13.4 m/s (30 mi/h) stops during a collision in a distance of 0.65 m. Estimate the average force that the air bag and seat belt exert on the dummy.
Sketch and translate We sketch and label the situation as shown below. choosing the crash test dummy as the system object. The positive x-direction will be in the direction of motion, and the origin will be at the position of the dummy at the start of the collision.
Simplify and diagram We model the dummy D as a point-like object and assume that the primary force exerted on the dummy while stopping is due to the air bag and seat belt’s \vec{F}_{\mathrm{A} \text { on } \mathrm{D}} , shown in the force diagram. We can ignore the downward gravitational force that Earth exerts on the dummy \vec{F}_{\text {E on D }} and the upward normal force that the car seat exerts on the dummy \vec{N}_{\text {S on } \mathrm{D}} since they balance and do not contribute to the acceleration.
Represent mathematically To determine the dummy’s acceleration, we use kinematics:
a_x=\frac{v_x^2-v_{0 x}^2}{2\left(x-x_0\right)}
Once we have the dummy’s acceleration, we apply the x-component form of Newton’s second law to find the force exerted by the air bag and seat belts on the dummy:
a_x=\frac{F_{\mathrm{A} \text { on } \mathrm{D} x}}{m_{\mathrm{D}}}=\frac{-F_{\mathrm{A} \text { on } \mathrm{D}}}{m_{\mathrm{D}}}=-\frac{F_{\mathrm{A} \text { on } \mathrm{D}}}{m_{\mathrm{D}}}
\Rightarrow F_{\mathrm{A} \text { on } \mathrm{D}}=-m_{\mathrm{D}} a_x
Solve and evaluate We know that v_{0 x}=+13.4 \mathrm{~m} / \mathrm{s} \text { and } v_x=0 (the dummy has stopped). The initial position
of the dummy is x_0=0 and the final position is x = 0.65 m. The acceleration of the dummy while in contact with the air bag and seat belt is:
a_x=\frac{0^2-(13.4 \mathrm{~m} / \mathrm{s})^2}{2(0.65 \mathrm{~m}-0 \mathrm{~m})}=-138 \mathrm{~m} / \mathrm{s}^2
Thus, the magnitude of the average force exerted by the air bag and seat belt on the dummy is
F_{\text {A on D }}=-(60 \mathrm{~kg})\left(-138 \mathrm{~m} / \mathrm{s}^2\right)=8300 \mathrm{~N}
This force [8300 N(1 lb/4.45 N) = 1900 lb] is almost 1 ton. Is this estimate reasonable? The magnitude is large, but experiments with crash test dummies in the real world are consistent with a force this large in magnitude, a very survivable collision.
Try it yourself: Find the acceleration of the dummy and the magnitude of the average force needed to stop the dummy if it is not belted, has no air bag, and stops in 0.1 m when hitting a hard surface.
Answer: -900 m/s² and 54,000 N.