So, Texas wants to raise the speed limit to 85 mph. What do I think? Well, to tell you the truth, I usually drive 5 mph under the speed limit. I change this driving habit when my wife is in the car. Then I go the speed limit.

But the real question (actually there are two big questions) is what about safety? This is actually a pretty tough question to answer. The problem is that collisions depend on so many things. If this is too difficult a question to answer, change it. That is the physicist way.

## Simplified Car Model

To explore the difference between crashing a car at 70 mph and 85 mph, I will use a model. This car doesn't have a crumple zone, it has a huge spring on the front. Here is a diagram.

Now, I am going to take this spring car and crash it into a fixed wall. When that happens, the spring will compress. There are two questions. First, how much does the spring compress? Second, what is the maximum acceleration of the car during this collision? I like to look at the acceleration because that is a good indication of possible injury.

## Work Energy

The work energy principle says that the work done on an object is equal to its change in energy. If I take the spring and car as my system, then there is no work done on it during the collision. The car will decrease in kinetic energy and increase in spring-potential energy. This can be written as:

Here I am calling the "1" position right before it hits the wall and the "2" position when it hit the wall and stops. This means that K_{2} will be zero (because it is stopped) and U_{1} will be zero because the spring is not compressed yet. The kinetic energy and spring potential can be written as:

For the spring potential energy, *k* is the spring constant. A higher *k* means a stiffer spring. Also, *s* is the distance the spring is compressed. Putting these expressions into the work-energy principle, I get:

That tells me how much the spring on the car is compressed. This would be like the amount of damage that was done to the car. Oh, I know a real car isn't just like a spring - but this model will give us something to work with.

## Force and Acceleration

What about the acceleration of the car as it crashes into the wall? Here is a force diagram for the car while it is crashing.

The two vertical forces (gravity and the road) clearly are not too important. They don't do work (because they are perpendicular to the motion) and even if they did, the two forces would cancel. What about the wall? Since the spring is compressed, it pushes on the wall. Forces are an interaction between two objects. This means that if the spring pushes on the wall, the wall has to push on the spring with the same force. I can write the magnitude of the force the wall exerts as: