Missile Hazard in a Collision

Question:

I failed physics and never took calculus and don't know anything about laws of motion but I do know that a kleenex box on the back dash of your car can kill you in an accident where there is sudden deceleration because it will hit you in the back of the head with enough force to do so.

Could you please explain how this happens in terms of acceleration-velocity-momentum-force etc because I would like to understand the science/physics behind it.

Answer:

I am going to work this out in metric units because we need to deal with the mass of an object and the English unit for mass is the slug, which is probably even less familiar to you than is the kilogram. Also the math works out easier in metric units. I will include the English units where they would help you picture what is going on. What I plan to do is find the force with which the box of kleenex would hit the skull of a passenger in the front seat and leave it to you to decide if that would be as deadly force.

Let's suppose that the kleenex box (with kleenex) has a mass of 0.25 kilogram, about 1/2 pound. If your car is going 100 kilometers per hour, about 60 mph, then the speed of the box before the collision will be the same. This works out to 100,000 meters per hour and since there are 3600 seconds in an hour the speed is 100,000/3600=28 meters per second.

Now let's think a bit about the details of the collision. Suppose the car hits an immovable object like a bridge abutment. The front bumper of the car stops almost instantly and the whole front end crumples. The force of that deceleration triggers the air bag which catches the passenger, who is still going 60 mph and brings him to a stop in about 1/10th of a second. The average force on an object in a collision is the change in momentum divided by the duration of the collision. The momentum is the mass multiplied times the speed. If the passenger had a mass of 70 kilograms, about 150 pounds, then the stopping force on the passenger would be about 70kg times 28m/s divided by 0.1s. This works out to about 19,600 Newtons or about 4400 pounds of force. Because the airbag covers a lot of the surface area of the passenger, this force is distributed over the chest, abdomen, arms and head, perhaps an area of 200 square inches. The force per square inch then about 22 psi. This is a huge jolt, but possibly survivable.

Now, getting back to the kleenex box, since it at the time of the collision is about 2 meters behind the passenger's head, and there is assumed to be nothing to slow the box down, it will take the box about 2 meters per second divided by 28 meters per second or 0.07 seconds to travel the distance to the pasenger's head. The airbag will pretty much have the passenger stopped by the time the box gets there so let's assume that the difference is speed between the box and the passenger is 25 m/s. The momentum of the box then is 25m/s times 0.25kg or 6.25 kgm/s. The box is going to crumple a bit when it hits the skull so lets allow 0.01 seconds for the skull to bring the box to a stop. Again the force is the momentum change divided by the duration of the collision or 6.26/.01=626Newtons. This is an average force during the collision of about 140 pounds.

Unlike the airbag, the kleenex box has sharp edges and corners. The area of the box involved in the collision will start out small and increase as the box crumples. Also the average force will be a lot less than the peak force because the box does not have the airbag's cushioning effect. If initially there is one square inch of the skull, exposed to 4 times the average stopping force then the skull would experience about 560 pounds per square inch. I suspect that this might break the skull and leave the brain to share in the work of stopping the kleenex box.