An Estimation of the Wind Speed Inside a Tornado

Here is a video from the recent devastating tornado that passed through the Dallas area.

The request was to find out how high these trailers went. I think a better question is to use the flying trailer as an estimate of the wind speed. Let me do this by making some crazy assumptions to build a basic model of force that wind pushes on something.

Air Force Model

Although I am starting with simple ideas, I won’t need a spherical cow for this calculation. Instead let me look at wind that is hitting a box at a 90° angle. Here is a picture.

I know that diagram isn’t all too helpful, but it is something to start with. Here I have a box that has an area A facing the wind that is moving with a speed v. Why would there even be a force from this wind? What if I think of the wind as a big box of air that collides with the trailer? Suppose that this air-box collides and stops (which wouldn’t happen). Using this air, I can think of the momentum principle to find the force. (I am just dealing with one dimension, so I will write these things as scalars instead of vectors)

If I just know the change in momentum of this “air box” and the time it takes to interact with the trailer, I can get an estimate for the force. First, I need the mass of this air. Let me call ρ the density of the air. Then the volume and the mass of this air will be:

In case you couldn’t tell, L is the length of this box and A is the area of the side. Here is the trick. Let me pick some time interval Δt. The longer the time, the longer the length of the side of the box. In general, the length L will be vΔt. This means the mass of this air will be (oh, v is the speed of the wind, right?)

Now I can get the change in momentum (since the final momentum is zero) and the force from the air.

The negative sign on there is because this is actually the force the trailer pushes on the air. Since forces are an interaction between two things, the force the air pushes on the trailer would be the opposite (so in the same direction as the wind). What about units? Does this have the same units as force? Well, density is (kg/m³) area is m² and the velocity-squared is m²/s². Multiplying all these together gives kg m/s² – which is the same as a Newton. This is good. Also, what happens as you increase either the density of the air, the windspeed or the surface area? An increase in any of these will increase the force. This makes sense.

Lifting a Trailer

Ok, how would you lift a trailer? Let me start with a super-simple case. Suppose the wind is blowing straight up and lifting the trailer. Also, like before, I will assume the air hits the trailer and just stops. Here is a diagram.

In order for this trailer to be moving up at a constant speed, the gravitational force and the air force (not THE Air Force) would have to have equal magnitudes. What is the mass of an empty trailer and how big are they? With a little searching, I found this answer. It lists the empty weight at 32,000 pounds with dimensions of 53′ x 8.5′ x 9.2′.   Close enough. This would give a weight of 1.4 x 10⁵ Newtons with an area (on the bottom) of 41.8 m². Setting the gravitational force equal to the air force, I can solve for the wind speed.

UPDATE: Thanks to a reader (Mickey), I see that I have used the weight of the entire trailer with the truck.  He suggests a value of up to 15,000 pounds is better (6.7 x 10⁴ Newtons).  Here is a site with that info.

I guess I should say that the density of the air is right around 1.2 kg/m³. Putting in these values, I get a wind speed of 165 m/s 36 m/s which is about 80 mph. The Wikipedia page on tornadoes says that extreme tornadoes can get over wind speed of 300 mph. Where does this leave my calculation? Well, remember that I made some estimates that clearly weren’t true. For instance, if I consider the mass of air to “bounce back”, this would double the change in momentum of the air and thus double the force on the trailer. Using this idea would decrease the necessary wind speed to 260 mph. Of course, in the real case the trailer is at an angle with the wind likely blowing horizontally. This means that only a component of the force from the wind is in the vertical direction.

Note: I also realized I made a mistake in my calculation above (which is now fixed).  For some reason, I used the weight as the mass.  This means I had an extra factor of 9.8 in there.  80 mph seems to be a good value.  Of course, this would be in an ideal upward wind case.  You would likely need a higher wind speed to get the trailer up from the side.  200 mph seems like a plausible value.

This was just a rough estimate to see how possible it would be to lift one of these trailers. Clearly it is possible and clearly these were pretty high winds.