Lastufka Labs - Reference

# Scale Kinematics

Part of the thrill of any miniature race is imagining that you are the driver, captain, pilot, etc.. To the extent that your design and execution of it in the construction of your vehicle is an extension of your skill, this is true. But if you really could be shrunk down to the size of your vehicle, what speed and time would you experience?

For a simpler illustration below, let's use a pinewood Grand Prix car. The methods shown below can be followed for any miniature vehicle. The example given here derives a scale for the entire race, not just one car. This approach might not be appropriate for all events. For instance, a Raingutter Reggata typically has many different classes of boats and ships represented. The scale of individual boats would vary dramatically. In fact, if scale speeds were calculated for all the boats after the race, it is very possible that the winner would have a lower scale speed!

## Setting The Scale

The key to determining scale information is to set a length for how long a Grand Prix car would be if it were a real full-sized car. To do this, we must answer the question, "How long is the typical car that contestants' Grand Prix cars model?". It wouldn't do just to use the car you modeled your racer after because everyone would have different scale speeds and times depending on their modeled cars. So what class of car does one pick? Perhaps your club models stock cars more than Formula I type cars, or maybe even dragsters.

A better way to determine the up-scale class is to look at the ratio of the length of the car to its width. For our small cars, the length to width ratio is 7 inches to 2 3/4 inches, that's about 2.5 as a fraction. This is also called the car's aspect ratio. So look for a class of real cars that has the same aspect ratio. If you want to match more details, you can also look at the ratio of length to wheel diameter; 7/1.1 = 6.4 for our down-scale cars. Let's call that the wheel ratio.

Here are some rough ratios I estimated from pictures and calculated from books:

Car Class Aspect Ratio Wheel Ratio Wheel Base (feet) Length (feet) Top Speed(mph)
Formula I 1.8 7.5 8.3 14 200
Pro Stock 2.6 6.5 8.3 14 190
Funny Cars 3 4.6 to 14 8.3 to 10.4 14-17 290
Dragsters 5 to 6 10 to 30 25 30 290
Pinewood Kit 2.5 6.4 0.092 to 0.48 0.58 12.27

Based on this table, a good pick that scales down to our car size reasonably well is Pro Stock. This class of cars has a length of about 14 feet. So, let's assume that our 7 inch pinewood cars have a scale length of 14 feet. That's a model scale-down of 1/24 (=0.58/14), or we can say our pinewood racers are 1/24th the size of the cars we are modeling. Note, if we use 14.5 feet as our scale length, since all Pro Stock cars are not the same length, the scale becomes 1/25! You can see that this method is not perfect.

## Scale Distance

Now the real fun begins. We know that the 7 inch length of our model cars is somehow equivalent to a typical length of Pro Stock class cars. If this is so, then every time our car rolls 7 inches, it went one car length which is 14 scale feet. If our car rolls 30 feet, it has gone 30*12 = 360 inches or 360/7 = 51.4 car lengths. That's 51.4*14 = 719.6 scale feet. Equivalently, knowing that the "real" car is 24 times the size of our pinewood model (the up-scale) we can write:

30 real feet x 24 = 720 scale feet.

So to get scale distance, you can either figure out how many car lengths your vehicle went and multiply by your vehicle's scale length, or just multiply the distance in feet your pinewood car went by its up-scale factor.

## Worksheet for Scale Distance

Real Distance(feet) Up-Scale Scale Distance (feet)
Example: 30 24 30x24 = 720

## Scale Time

When scaling up, the lengths increase a lot. How about time? The typical Grand Prix race takes only 3 seconds. Does it increase also? If our only principle to go by was to compare actual times over scale distances, we would be in trouble. As technology improved, cars went faster. Just over 60 miles an hour was tops in 1900. Dragsters are now approaching 300 mph. So we need a different way to determine whether the time changes when we scale up.

This is not an easy question to answer, but since any experiment we might come up with to test our reasoning about a method would be run in "real" time, we have to consider that time must remain the same since at present we can't speed it up or slow it down.

On the other hand, if we made a very small version of a car, say the size of a hydrogen nucleus, and measured it's scale speed using its scale distance and "real" time, it could easily exceed the speed of light! As far as we know, even such small models must not do that. So it seems if we let things get too small, we need to use relativistic physics rather than Newtonian physics as we always have.

To avoid that complication, we'll assume (safely) that we won't have speeds that are nearly high enough to warrant relativistic considerations. We'll take time to be the same at all our scales of interest.

## Scale Speed

If time is the same in our real miniature race as in our fantasy scaled-up race, then scale speed is scale distance divided by our real time. For the example above and the typical 3 second time that's 720/3 = 240 ft/s. For miles per hour we multiply by 60 seconds to the minute and by 60 minutes to the hour and divide by 5280 feet to the mile. That's a factor of 60*60/5280 = 0.6818 mile-second/foot-hour. 240*0.6818 = 164 mi/hr or 164 mph as we're used to seeing it.

As you can see from the table above, that's not far from the performance of a real stock car. To reach a scale speed of 190 mph, a pinewood car would have to master a thirty foot track in 2.5837 seconds.

To get scale speed

## Worksheet for Scale Speed

Scale Distance(feet) Real Time(seconds) ft/s to mph Scale Speed(feet)
Example: 720 3 0.6818 0.6818*720/3 = 164

## Other Scale Parameters

Just as we determined scale speed, we could define scale acceleration by looking at the change in scale velocity over time as long as it doesn't go relativistic. Scale volume, would be scale height times scale width times scale length.

What about scale weight and other interesting forces (dynamics)? For those, engineers look at ratios again. A particularly important one in aircraft and ship modeling is the the Reynolds number. A few physical quantities appear in the equation for this number. To scale them to one another as we have done for lengths and speeds, it turns out that they must take on values that keep the Reynolds number at the same value at both scales.

For our purposes, assuming that the weight per cubic inch is the same for large and small helps answer some of these questions. For example, if our example car weighs 5 ounces and measures, on average, 7 inches by 2 inches by 0.5 inches, then it's volume is 7 cubic inches and its scale volume would be 56 ft3. The weight density of the small car is 5 oz/7 in3 = 0.7142 oz/in3 or 77.13 lb/ft3 (lb is pounds). If the scaled-up car has the same weight density, it would weigh 77.13*56 = 4,319.5 pounds. That's a heavy car considering that a Formula 1 car weighs in at only 1,270 pounds plus about 350 pounds for fuel. Perhaps it's not so surprising that it's so heavy, real cars have lots of air filled cavities in them and use light metals and plastics that are not much more dense than pinewood.

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