76 Safety Technology

Bounce is Bad

If the skull impacts a hard surface, but neither the skull or surface are significantly deformed, then the skull will bounce back. This type of collision is not perfectly inelastic because the skull did not stick to the surface. In this case the relative impact speed between the still-forward moving brain and the backward moving skull is nearly twice what it would have been if the skull had simply stopped.

A human skull moving forward and impacting a solid wall. A cutaway of the skull shows the brain inside moving forward, impacting the front of the skull, then moving backward and impacting the back of the skull. Injured areas on the front (frontal lobe) and rear (occipital lobe) of the brain are highlighted.
The alternating accelerations of the skull and the inertia of the brain combine to cause impacts on opposing sides of the brain during a coup contrecoup injury. Image credit: Contrecoup by Patrick J. Lynch, medical illustrator via Wikimedia Commons

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After the bounce, as the neck causes the skull to slow down on the way backward, the inertia of the brain may lead to a second impact on the back of the brain, as illustrated in the previous image. Aside from an additional brain tissue injury, the combined swelling of the two opposed injuries will put amplify the pressure on the brain and increase the likelihood of permanent injury. This type of injury is known as a Coup Countrecoup , or translated from French by Google Translate, blow, counter blow.

Air Bags

Everyday Example: Airbags

Check out this video of crash-testing with and without airbags.

https://youtu.be/NACA1W2A5Wk

During the crash the driver’s head starts out with the same velocity  as the car. The airbag increases the time it takes for the head to come to rest, which will decrease the force on the head. In order make sure the head does actually come to rest instead of bouncing the airbag has vents that allow air to be pushed out during impact so  it deflates instead of staying completely full and keeping its shape.

The impulse-momentum theorem introduced in the previous chapter will help us to analyze technology designed to minimize the forces on your the body:

(1)   \begin{equation*} \bold{ p_f}-\bold{p_i} =\Delta t \bold{F_{ave}} \end{equation*}

If we divide over the time, we see that there are two options for reducing the force on the body during a collision; you can maximize collision time, or minimize the change in momentum, or both:

(2)   \begin{equation*} \frac{\bold{p_f}-\bold{p_i}}{\Delta t } =\bold{F_{ave}} \end{equation*}

Through our analysis of locomotion, falling, and landing we have already examined strategies for increasing the collision time, but how do we actually reduce change in momentum you experience during a collision? Prevent the bounce! Let’s examine why the bounce is bad using an the airbag example.  First we write the momentum of the head in terms of its mass and velocity.

(3)   \begin{equation*} \frac{m\bold{v_f}-m\bold{v_i}}{\Delta t } =\bold{F_{ave}} \end{equation*}

If the collision is perfectly inelastic so the head does not bounce at all, then final velocity is zero:

(4)   \begin{equation*} -m\bold{v_i}}{\Delta t } =\bold{F_{ave}} \end{equation*}

We see that the force on the head depends on the head mass, the initial velocity, and the collision time.

If instead the head bounces back at the same speed it went in (image the airbag was a rubber ball), then the final velocity is just equal and opposite (negative) the initial velocity:

(5)   \begin{equation*} \frac{m(-\bold{v_i})-m\bold{v_i}}{\Delta t } =\bold{F_{ave}} \end{equation*}

The left side can be combined:

(6)   \begin{equation*} \frac{-2m\bold{v_i}}{\Delta t } =\bold{F_{ave}} \end{equation*}

That is twice the force compared to when the head just came to rest! If the head were to bounce back, but not at full speed, then the force would still be greater than the perfectly inelastic case, but not twice as great.

Reinforcement Exercises

  1. Contrecoup by Patrick J. Lynch, medical illustrator [CC BY 2.5 (https://creativecommons.org/licenses/by/2.5), GFDL (http://www.gnu.org/copyleft/fdl.html) or CC-BY-SA-3.0 (http://creativecommons.org/licenses/by-sa/3.0/)], via Wikimedia Commons ↵
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Body Physics: Motion to Metabolism Copyright © by Lawrence Davis is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

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