Scale Airplanes and Scale Speed

Scale models are not easy. I, of course, am reluctant to pass on any building and craftsmanship tips to anyone. A look at my picture on the page you used to get here will explain why. I don't need to make enemies. But the problem with true scale is something to think about.

Suppose you had a cube of water 1-foot on a side. It weighs 64 pounds. (Using approximate round numbers here to make a point). You want to make a 1/4-scale model of it, that is a cube of water 3 inches on a side. It will weigh 1 pound.

The general rule is, if you are shrinking every dimension by a scale factor, s, the volume changes by a factor of s³. If the material stays the same the weight will scale by the same amount. The areas change by a factor of s². In the cube of water example above s=0.25 and s³=0.015625.

So try this on a reasonable size airplane. The C-47 is a well-known aircraft and, in fact, is modeled to 1/14 scale by Top Flite. The original and the model had or have the following (approximate) characteristics:

The takeoff weight allows for crew of four and 165 gallons of fuel. Nothing else. From my experience the C-47 I flew in was heavier than that. The cruising speed is sometimes listed as high as "185 mph", which seems unlikely to me, but then our C-47's were pretty war weary.

So, if you took a real C-47 and shrunk it to a 1/14 scale by taking each group of 14³ atoms of aluminum and replacing them by one atom, and so-on for each element, you would have a possible winner in static judging. Here s =0.0714.., s²= 0.00510.., and s³=0.000364... I leave all the remaining decimals off, which you can supply with a pocket calculator.

OK, so the real scale model is a little on the heavy side, but what about the speed? Where did that come from? This is the key to the problem.

Clearly, you would like to have the model fly at a speed that made it look right in flight. A speed that would have the model cover its own length in the same time as the full size aircraft. When the model flew past you at a distance of 100 feet it would look like a full size C-47 at 1400 feet, and move at the same number of degrees per second.

Nature doesn't work that way. About the only thing you can do to keep the illusion is to make the model much lighter than the constant density scale down. That is possible with some of the materials available today and the craftsmen that take up this pursuit. But it ALWAYS works out better if you build a model that is 1/4 or 1/3 scale. The closer the scale factor is to 1 the easier it is to match the appearance of scale flight.

The same is true, of course, if you, for instance, tried to scale up a small rubber-powered model to 14X. A 1-foot span model becomes 14 feet and the glide speed of 3 ft/sec becomes 42, nearly 30 mph.

As a scientific curiosity, if you scale down an airplane the density (which was assumed constant in the C-47 shrink) has to go down with the square root of the scale factor if the speed scales like we want it to. Then the angle of attack will be the same at the same flight condition and the speed will scale with the scale factor. For the scale factor of 1/14 this would mean the density of the model would have to be about 26.7% the density of the original aircraft. For a 1/3 scale the density has to be 58%, a lot easier to come close to.

If you want to convince yourself, try the exercise above using a really big airplane; something like a 747, Hughes HK-1 (AKA "Spruce Goose"), or a KB50 scaled down to 6 foot span.