The Physics/Energy of a High-Speed Crash

Physics of a High-Speed Crash: 70 MPH vs. 85 MPH

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 K₂ will be zero (because it is stopped) and U₁ 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:

The more the spring is compressed, the greater the horizontal force on the car and thus the greater the acceleration of the people inside. So, the greatest acceleration will be:

And using the value for the maximum compression above, I get:

So what does this mean? It means that if I increase the initial speed, the max acceleration on impact increases by the same factor.

How About Some Values

I think I can do this without picking a mass of the car. Suppose that I have a car going 70 mph (31 m/s) and it crashes into a wall with a spring compression of 1 meter (I just randomly picked that). What would the value for m/k be?

Now I can use that for the maximum acceleration during a collision. Here are the values for 70 mph (31 m/s) and 85 mph (38 m/s)

Ok, I am happy. First, this is the acceleration at the maximum compression for a spring. However, my special spring doesn’t bounce back. Bouncing back would have a much greater acceleration than just stopping (because of the change in direction of velocity). But I guess it stops at that instant, so maybe that isn’t so bad.

The other problem is that the useful acceleration data really needs a time. A human can withstand super high accelerations as long as the time interval is short enough. So, what is the acceleration as a function of time? Acceleration depends on position, but position depends on velocity and velocity depends on acceleration. How about a quick numerical plot? First, this is the velocity of the car as it collides.

And here is the acceleration (as a function of time) for the car:

How bad is this acceleration? This is my favorite table that used to be on Wikipedia’s g-force page.

This says that if you are driving and crash into a wall, you would accelerate “eyeballs out” and could take about 28 g’s for less that 0.01 seconds. This is bad. Looking at the above graph, you would be over 28 g’s for about 0.04 seconds. Note to self. Don’t crash your car into a wall if you are going 70 mph even if the car has a huge spring on it.

UPDATE: I was wrong (as pointed out in the comments).  The table above says that the time is in minutes, not seconds.  Dooh!  Anyway, looking again at Wikipedia’s human tolerance page – it lists 50 g’s as pretty much fatal.  So, this is still bad.

Here is a plot of accelerations for different starting velocities.

The faster you go, the worse the acceleration when you crash into that wall (which you totally should not do). But a couple of key points:

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• This is just a model using a spring to simulate the crushing of the car.
• The above graph shows the acceleration of the car. The person inside would have a different acceleration. Just imagine an air bag inside. The person would actually move forward more than the car (and decrease the acceleration). The person is not rigidly attached to car (at least I hope not).
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• Driving is dangerous. Driving is especially dangerous if there are walls in the road. I would just avoid any road like this. 
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