Collision safety design (crumple zones, airbags) Knowledge-Based Study

Study notes

1. In any collision, an object experiences a change in momentum given by $\Delta p = m\Delta v$. According to the impulse-momentum theorem, this momentum change is equal to the impulse applied by the net external force:

   $$J = F_{\text{avg}} \Delta t = \Delta p$$

   When a vehicle of mass $m$ traveling at an initial velocity comes to a complete stop, the total change in momentum $\Delta p$ is fixed entirely by its initial mass and velocity. Therefore, the product of the average impact force $F_{\text{avg}}$ and the collision duration $\Delta t$ remains constant.

   Vehicle structures are designed with deliberate structural deformability known as crumple zones. Rather than remaining completely rigid during an impact, the front and rear zones of the car collapse in a controlled manner.

   • Increased Impact Duration: As structural components deform plastically, the time $\Delta t$ required for the vehicle cabin to decelerate to rest increases significantly.
   • Reduced Peak Force: Because $F_{\text{avg}} = \frac{\Delta p}{\Delta t}$, extending the duration of the momentum change drastically reduces the average force transmitted to the passenger compartment.
   • Energy Dissipation: Controlled plastic deformation also converts mechanical kinetic energy into internal thermal energy and material deformation rather than transferring it directly to vehicle occupants.

   Even if the car body slows down over an extended period, occupants continue moving at the initial velocity until acted upon by an external net force, in accordance with Newton's first law of motion.

   • Seatbelts: Restrain passengers within the protected cabin space and stretch slightly under high tension to prevent immediate collision with the dashboard while moderately extending deceleration time.
   • Airbags: Rapidly deploy to provide a compressible cushion between the occupant and rigid interior surfaces. As the occupant compresses the deflating bag, the deceleration time $\Delta t$ of the head and torso increases further.
   • Pressure Distribution: In addition to reducing $F_{\text{avg}}$ through increased collision duration, airbags distribute the remaining force over a much larger surface area, greatly minimizing localized impact pressure on vital organs.

   Consider an occupant with mass $m = 70\text{ kg}$ stopping from an initial speed of $20\text{ m/s}$. The required change in momentum is:

   $$\Delta p = m(0 - v_0) = -1400\text{ kg}\cdot\text{m/s}$$

   If the occupant strikes an unpadded, rigid steering wheel and stops within $\Delta t = 0.01\text{ s}$:

   $$|F_{\text{avg}}| = \frac{1400\text{ kg}\cdot\text{m/s}}{0.01\text{ s}} = 140\,000\text{ N}$$

   If the occupant is cushioned by a combination of a seatbelt and an airbag that extends stopping time to $\Delta t = 0.14\text{ s}$:

   $$|F_{\text{avg}}| = \frac{1400\text{ kg}\cdot\text{m/s}}{0.14\text{ s}} = 10\,000\text{ N}$$

   By lengthening the collision duration by a factor of 14, safety devices lower the net impact force to a survivable threshold while handling the exact same change in momentum.