## Hidden Nasties

When we go to full scale… who knows what’s going to happen?
Tori Belacci

Well, I’ll mostly ignore the gross biology stuff from this episode, as it’s not my area of expertise.  I definitely prefer to pay attention to the destruction of automobiles.

I enjoy it even more when I get to talk about the Buckingham Pi Theorem.  Non-dimensional numbers are my favorite kind of numbers.

“But Ryan,” you say, “what’s all of this non-dimensional non-sense?”

Well, after their first test failed, you’ll kindly recall that Tori did some small scale tests with a 1:12 model of the Lamborghini.  However, instead of using a model that weighed 1/12 of the actual car, they went all the way down to two pounds.  And they scaled 50 mph down to 14 mph.  Where did those numbers come from?

This is where non-dimensional numbers come in.  If an event has the same non-dimensional numbers as another situation, they two situations should be indistiguishable.  This is the foundation of scale modeling.  If you can reproduce the relevant non-dimensional numbers, a small scale test should replicate a large scale test, only much cheaper and maybe safer.

For more details, I’ll refer you to the Wikipedia articles on dimensionless numbers and the Buckingham Pi Theorem.

Okay, so let’s think about the important characteristics of the car’s motion and interaction with the water.  I’ll go ahead and list what I would consider the five most relevant physical parameters of the problem:

• $L_{c}$ – length of the car
• $V_{c}$ – velocity of the car
• g – gravitational acceleration
• $\rho_{liquid}$ – density of water (or whatever liquid we’re trying to skip)
• $m_{c}$ – mass of the car (assume for simplicity, it’s distributed consistently throughout the car)

We expect there to be a couple of dimensionless numbers: $\frac{V_{c}^{2}}{L_{c} g}$ (square of the Froude number) and $\frac{m_{c}}{\rho_{liquid}L_{c}^{3}}$ (sort of a specific gravity). We want them to be the same in both the large and small scale tests.  There are a couple things we’re stuck with.  Since the car is scaled 1:12, it is, let’s say 1 ft (0.3 m) long instead of 12 ft long.  And gravity isn’t going anywhere, so we’re stuck with $g=9.81\:m/s^{2}$.  Maybe we could use a different liquid, but we’ll stick with water for now, so $\rho_{liquid}$. So that only leaves our new speed and new mass  (note, 50 mph = 22 m/s and 2600 lbm = 1200 kg):

$\left(\frac{(22 \: m/s)^{2}}{(3.6 \: m)(9.81\: m/s)}\right)_{full\: scale}$ = $\left(\frac{V_{c}^{2}}{(0.3 \: m)(9.81\: m/s)}\right)_{small\: scale}$

and

$\left(\frac{1200\: kg}{(1000\: kg/m^{3})(3.6\: m)^{3}}\right)_{full\: scale}$ = $\left(\frac{m_{c}}{(1000\: kg/m^{3})(0.3\: m)^{3}}\right)_{small\: scale}$

Solving for the small-scale values car gives $V_{c} = 6.4 \:m/s =14 \:mph$ and $m_{c}=0.7\:kg=1.5\: lbs\approx 2\:lbs$.  What do you know?  That was the velocity and mass they used!  That means they used science!  And since they had no explanation of why they did what they did, they left me something to talk about.  FTW!

Now, I must admit, I was slightly surprised/impressed that the team got as much right (mass/speed) as they did.  But considering they were in the business of special effects, it is a no-brainer that they know these ins and outs.  If you want to make the filming of a miniature disaster look like a full scale one (before the miracles of CGI came along), you better know a thing or two about dimensional analysis.

On the other hand, there are a few more potentially important variables that go into the physics of this particular problem, when we talk about the car skipping on the water.  If we included the viscosity and surface tension of the water, or air resistance, etc, we would find it almost impossible to duplicate all of the dimensionless numbers in the small-scale (it would have to involve some exotic liquids and gases).  But I think we hit the most important numbers for this situation, which is probably why the small-scale tests looked so similar to the full-sized tests.