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This week's CR4 Challenge Question:
We know that air bags save lives, and we all have them in our cars. We know that they are fast to open in case of an accident, they are very soft, and they absorb energy. Now, why do they save lives? Explain it in terms of physics.
And the Answer Is...(June 24, 2008: 8:33 AM EST)
To understand the reason air bags save lives, we must look at the concepts of impulse and momentum. In an accident, or when you stop the car softly or abruptly, what you really want to do is to change the speed of an object of mass m from v1 to v0 (where, in the case of an accident v0 = 0). In other words what you want to do is to change the momentum of the object according to the equation, Δp = mv1 - mv0 where p = mv is the equation for momentum. In the case of an accident, where the vehicle stops completely (v0 = 0), the change in momentum is simply, Δp = mv1 where v1 is the velocity of the vehicle at which the accident takes place.
We can derive an equation for the change in momentum by re-arranging Newton's Second Law of motion. The second law can be stated as

Where a is the acceleration of the object, m is its mass, and t is the interval of time during which the force is being applied to the object.
If we multiply t by both sides of the above equation we get
Ft = mΔv = Δp (1)
The quantity Ft is defined as the impulse, I. As we can see from Eq. (1), the impulse is equal to the change in momentum, or
Impulse = change in momentum
I = Ft = Δp
In the equation given above F is the force that is needed to change the momentum of the object an amount Δp. The time t is the time interval that this force should be applied. In the case of an accident, as stated above, this force can be written as
(2)
We can see from Eq. (2) that the force needed to stop the vehicle in an accident is inversely proportional to the time that the force is applied. It is clear that the longer the time interval the force F is applied, the smaller will be this force.
During an accident between two vehicles, each vehicle stops abruptly and each car exerts a force F onto the other vehicle. Because the time it takes to stop the vehicles is very small (almost zero), according to Eq. (2) the force is enormous. This is the reason the vehicles are normally completely destroyed. The person inside the vehicle that moves at speed at the moment of the accident will have a speed equal to zero after the accident. In order to change the momentum of the person inside the car from v1 to zero the same force F must be applied to the person during a time interval t by the dash (or any other surface) of the car.
Now, if the person hits the dash of the vehicle, and because the dash is sturdy, the time it takes to stop the person movement is extremely short. This means that the force exerted by the dash is extremely high. You well know happens in a case like this.
If, on the other hand, an air bag opens at the moment of the accident, the person inside the car will hit the air bag instead of the dash of the car. The air bag will exert a force on the person, but (this is the important part) during a longer interval of time. The bag will deform (unlike the dash) and it will hold the head (or the body) of the person for longer period of time during which a force is being applied to the person to stop the movement. According to Eq. (2) the force will be smaller. By receiving a small force to the head, the person survives.
So, air bags save lives because they extend the time period during which a force changes the momentum of your body.
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