It's not even summer yet, but it's hot enough in Minnesota to cause a highway to buckle. Many cars flew right over it. It looks dangerous. I hope there aren't any injuries, but I'm sure there will be cars with suspension problems.

Now for some physics.

Why does the road expand when hot?

When you think about solid materials, it's useful to model them as a bunch of tiny masses connected by springs, like this:

This is a GIF from a VPython program Bruce Sherwood created. It models the motion of 27 masses joined with springs. You can run the program and see the code at Glowscript.org.

But what about temperature? In this ball-spring model, a solid at a higher temperature would have the balls oscillating at greater speed. Of course, solid matter isn't actually like this, but the model works quite nicely. It explains how a solid exerts contact forces with other objects and how to estimate the speed of sound in a solid.

So how does this explain what happens when a solid heats? Really, if you use springs between the masses, increasing the temperature of a ball-spring model solid would make it oscillate more but not expand. The trick comes from a different potential energy for the springs in this model. If you use a non-symmetrical potential, then the average position of a ball in a solid will increase as it increases in energy. Here is a sketch of what that looks like.

This is from a previous post examining the physics of expanding solids. But in the end, a road will expand as it heats because the increased energy produces greater separation between atoms. If a road lacks room to expand, it buckles.

Treating the cars like projectile motion

When an object has a motion due to gravitational force alone, it's called projectile motion. When a car hits this buckle in the highway, it is launched with some initial velocity and at some angle above the horizontal. Once airborne, the only force acting on it is gravity (I'll ignore air resistance) such that it is mostly projectile motion.

Assuming the car starts and ends its jump at the same level, what value would you have for the launch angle? Let's look at this as a physics problem. I'll skip some details, but you can find many examples in my previous posts.

The main trick to solving projectile motion problems is to realize that you can treat the x-motion and y-motion as separate problems. Time is the connection between these two 1-D motion problems. The time required to move in the x-direction is the same as in the y-direction. Let's start with the x-direction. The acceleration in the x-direction is zero (since there are no x-direction forces) and the x-velocity can be found as the horizontal component of the velocity.

Just a note, that second equation comes from the definition of average velocity. Of course I can't solve for θ since I don't know the time it takes for this motion (even though I could estimate both the launch speed and the distance traveled). The next step is to examine the y-motion and solve for the time. In the y-direction I know the starting and ending position (both zero) as well as the acceleration.

Next, I can substitute this expression for t into the x-direction equation and then solve for θ.

In order to solve for θ, we need a trig identity. This gives the following expression for θ.

That's it. I simply need to insert estimates for the launch speed and the distance the car jumps. Here are my wild guesses. First, I'll say the car is going 60 mph (26.8 m/s). Second, I'll use a jump distance of 4 meters. This gives a launch angle of about 1 degree. Note that this is the launch angle, not the angle of the buckle. When the car hits the bump, the shocks and stuff compress, resulting in a different launch angle. Still, that angle isn't too much. It doesn't take much to get these cars airborne.

If you want homework, you could do this calculation starting with the "air time" that the car is aloft.

Looking at traffic

Finally, watch the video and notice the cars that encounter the bump. Each driver faces three choices:

Maintain speed and jump that thing. Yeeee-haaawww!

Slow down and approach it as a hazard or speed bump. This probably is the wisest choice.

Change lanes.

But what happens when a car slows down? The cars behind it must slow down. This can create some interesting situations—well, interesting to me but not to you if you're stuck in traffic. Personally, I like the way the Bill Beaty describes traffic in terms of waves.

And this cool video shows cars driving in a circle (so it's like infinite traffic).

Notice that when one car slows it can cause a traffic jam of sorts. You know what would be really neat? Making a simulation of cars in traffic and giving each some basic rules like, "stay 3 meters behind the car in front of you." Then add a bit of randomness and some delayed reactions to see if you get a traffic jam. I think I might try this.