In the last episode of MythBusters, Adam and Jamie wanted to test something that Jamie had said earlier:

"Two cars crashing head on at 50 mph is the same as one car crashing into a wall at 100 mph"

Jamie was wrong, but that is not what I am going to talk about. Instead, I am going to talk about Adam's small scale test of this situation. Really, it was a nice set up. Basically, he wanted to collide something into a wall at one speed and then double that speed. Then he was going to collide two things together at the lower speed. He had a cool way of measuring the collision. He put a piece of clay between two masses. When the object collided with something, the clay would get smashed and he could measure how smashed it got. Here is a simple diagram of his apparatus.

Where would you have to put the first object so that the second position is twice as fast?

## How fast as a function of angle?

Since these objects are not moving in straight lines, they actually don't have constant accelerations. I know the distance they travel though, so that hints that I should use the work-energy principle. In short, this says that if I look at the mass plus the Earth as my system, there will be no change in total energy. (Gravity won't do any work because it is an internal force to my system). This means:

Going from Position 1 to the bottom (which I will call position 0), I can write this change in energy equation. For simplicity, I will call the bottom of the motion the zero gravitational potential energy. This gives:

Notice that the mass cancels - that is a good thing. Also, in order to double the velocity at the bottom, y_{2} would have to be 4 times higher than y_{1}. Or, if you want to work backwards. Suppose you put the one mass at the horizontal position where position 2 is. What angle would position 1 be at? Here is the expression for the velocity from position 2:

Now, how high would y_{1} be so that v_{0-1} is half of that value?

If height is determined by an angle measured from the vertical axis, then (and the radius of the circle is R):

Ok, I admit. I first looked at the video and thought Adam had put the position 1 at 45 degrees (which wouldn't even be half the height). I was wrong. Here is a screen shot of the set up.

See the mark that says "49"? That must be the angle as measured from the top. Hats off to you, MythBusters.

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So what is it you're trying to say exactly?

rijkswaanvijand, Mythbusters frequently gets criticism from various people for not getting the physics right, either through misuse of terms (force, velocity, and acceleration, mostly) or through simple mistakes like they avoided above. Most people would have simply put the 1x height at 45 degrees instead of the correct one as calculated. It looks like Rhett expected them to as well, and was pleasantly surprised when they didn't.

I'm just glad to see this because I noticed the "49" while watching, but didn't understand why it was done that way. Now I know.

Rhett, what took you so long? That episode aired 17 hours ago!! (haha). I was thinking exactly of the 45 degree angle, and your impending dissection of the physics, as I watched it last night.

It makes me wonder why they don't bother to have a physics consultant on the show's production team -- and if Grant is really an electrical engineering grad, then shouldn't he be catching some of this too?! (or, as a civil engineer myself, I should realize that the EE's didn't really spend too much time focusing on kinetics and kinematics!).

Ok, I'm going to ask a stupid question and really display my ignorance. But what the heck.

When Adam did the clay test with 2 objects at speed X, he put one piece of clay in each object. Each clay piece showed the same compression as 1 clay object hitting a wall at X speed. That made sense to me even thought I didn't know the formulas.

But I yelled out to my wife that he did the test wrong. If he had put 1 piece of clay between the two objects, wouldn't that have compressed as much as the clay in the 2X speed test? That piece of clay would have absorbed an X mile an hour crash on each side. Wouldn't that be the same as one 2X crash?

That's how I've always imagined this thought experiment.

If I'm standing still in the middle of an intersection, wouldn't getting hit by one car traveling 50 miles an hour be equivalent to getting sandwiched between 2 cars going 25? Assuming that I'm super-glued in the spot and can't move from the impact.

Or am I just ignorant? Hopefully I'm not stupid as well.

@Bob,

I was going to talk about the clay, but I decided not to include it. In short, you are right in that you need to think about energy. If the clay exerted a constant force while it was being compressed, then the compression amount would be proportional to the amount of work the clay did on the mass.

@Bob "If I'm standing still in the middle of an intersection, wouldn't getting hit by one car traveling 50 miles an hour be equivalent to getting sandwiched between 2 cars going 25? Assuming that I'm super-glued in the spot and can't move from the impact."

Superglued to the spot might be more like being sandwiched against a bridge abutment. The trick in thinking about how much you'd get squished is to think about the energy you'd have to dissipate. The energy in 2 25mph cars hitting you, one on either side of you, would only be 1/2 of the kinetic energy (E=.5mv^2) you'd have to absorb from single 50mph car squishing you against wall.

I have to admit I guessed the outcome wrong too. I had figured double the energy, energy goes by speed squared, same as 70 miles an hour into the wall. But the doubled energy is being distributed into two cars, so it doesn't make a difference.

I'm confused. Regarding:

"Two cars crashing head on at 50 mph is the same as one car crashing into a wall at 100 mph"

Why is that wrong?

I mean, sure, it'll be wrong if you stick a compressible object of non-negligible mass at 0 mph between the colliding objects (maybe like a lump of clay?). But that hardly seems to be what's implied by the original statement.

If there's nothing but two identical cars and you're measuring the damage done to the car, shouldn't the two situations be the same? (Assuming an incompressible wall, neglecting friction with the ground, yadda yadda yadda.)

Whoops. My bad.

Somehow I parsed the original statement as "Two cars crashing head on at 50 mph each (in opposite directions) is the same as one car crashing into an incompressible wall at 50 mph."

Nevermind.

Dude, what is up with the tiny type?

@Art,

Oops - I missed a close tag for my subscript - I fixed.

Thanks for pointing that out.

"and if Grant is really an electrical engineering grad, then shouldn't he be catching some of this too?!"

Adam/Jaime work at M5 - Jaime's original work shop. Kari/Tory/Grant work down the street at M7 (another building) which would make it difficult to know the minutiae of what Jaime/Adam are doing. Plus, film schedules seem to be tight enough that he doesn't have time to worry about them either.

BUT WAIT!!!

I watched the episode and did the same analysis with the 49 vs 41 degrees. But I also did a screenshot of the two-pendulum (head-on) collision and realized that Adam measured 49 degrees from the bottom, not the top. (I simply held the corner of a piece of paper up to the vertex, and the combined angle was greater than 90 degrees, so each must have been greater than 41, i.e. 49.

So, despite getting everything calculated right, he measured the angle from the wrong side and lifted the pendulum to 49 degrees from the vertical instead of only 41 degrees.

Oops.

so "Two cars crashing head on at 50 mph is the same as one car crashing into a wall at 100 mph" is a correct statement or not?