Silence is the virtue of fools. -Sir Francis Bacon
Question #90080 posted on 07/15/2017 2:50 p.m.
Q:

Dear 100 Hour Board,

So I was playing with toy cars with my son Twist yesterday and a question occurred to me. When I take a toy car in my hand and zoom it as fast as I can through my arm's range of motion, how fast is it actually going? Does it travel that very short distance as fast as a real car would or much slower?

-Inverse Insomniac

A:

Dear Inverse,

Quite serendipitously, I was visiting my parents in Rubikland when this question came in, so I had access to toy cars with which to conduct my research. While looking for my old Hot Wheels, I also found these old toys, which I thought I'd share with y'all:

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First, we have Hoth Luke Skywalker and Endor Han Solo, along with my A-Wing calculator. That's right: an A-Wing calculator. Can you conceive anything more nerdy awesome? I didn't think so.

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Here we have a couple of Bulbasaurs. I was going to take the plush one with me when I went back to Provo, but then I realized that it is Cadet Rubik's and my plush was a Squirtle. I couldn't find the Squirtle. I was sad.

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Mère Rubik calls this set "Cool Tools." I have no idea if that's what the actual brand of toys was called or if she just thought they were really cool. I suppose we'll never know.

Lastly, we have one of my most favorite toys...

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...the Thing. I do not know what this thing is. It is made of solid steel and is fairly heavy. I remember that I picked it up one day and thought it was cool, but that's all I know about it. When I had it out for this answer Père Rubik saw it and said "Oh, hey, it's that thing." I asked him what it was. He said that he didn't know, but he had randomly received it from someone long ago. So we still have no idea what the Thing is. If a reader could tell me, with convincing proof, what exactly the Thing is, they would have solved a lifelong mystery for me and I would certainly owe them ice cream.

Alrighty. On to the cars:

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These cars were ultimately deemed unusable for the experiment, but I have fond memories of them and wanted to share them with y'all. The dark green racer (second from left) is from 1987! The lime-green racer next to it (in the middle) is from 1982! These cars are 30 years old! It's incredible. 

Here are the cars I used for my test:

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From left to right, here are the models of the cars:

Car #1: Ferrari F50

Car #2: Ferrari 156

Car #3: Dodge Viper GTS

Car #4: Renault Formula One Racer (I couldn't find any more specifics on exactly what kind of Renault F1 it is)

Car #5: BMW Z3

Now, how do we tell how fast they're going?

First, we measure my arm, from fingertip to shoulder. It's about 30 inches long. Next, I measure how high off the ground my arm is when I'm sitting on the ground, ready to race. That's about 25 inches. Using the Pythagorean Theorem, I find that the horizontal distance from my shoulder to the car in my hand is about 16.5 inches.

This last measurement is very important. For the test, I'll be moving the cars back and forth in a quarter circle, from when my arm is straight out to my side to where it's pointing straight in front of me. The distance it will travel is the radius of that circle (16.5 inches), multiplied by Pi, and divided by two. That distance ended up being right around 26 inches.

Now, for the actual test: I need to find out how long it takes me to drive the car through the quarter circle. This would be very hard to time if I were just doing it once, so I decided I would drive the cars back and forth as fast as I could for 30 seconds so as to get a nice sample size from which I could calculate the time for a single run. Here's how each car did:

Car #1: 74 "laps"

Car #2: 80 "laps"

Car #3: 84 "laps"

Car #4: 88 "laps"

Car #5: 89 "laps"

For the most part, I thought it was a pretty good method, but I did identify a couple of issues. For one, as you can see, the number of "laps" each car ran in the 30 seconds increased every time. I was actually worried that the opposite would happen; I figured that my arm would get tired and that I wouldn't be able to move it as fast, so in between each run (besides the last two) I took a break to work out calculations and eat hummus with pita chips. Apparently, that worry was unfounded. The only other issue was that a couple of times during the testing I had to start the runs over because I was so excited making the cars go vroom Vroom VROOM that I lost count of how many laps they'd gone.

To even things out, I averaged the number of "laps" between the five cars, and found that it was 83. Now, if a car can travel that distance 83 times in 30 seconds, then it can traverse it one time in 30/83 = 0.361 seconds. If we divide our distance of 26 inches by this time, we find that the car is traveling 72.068 inches per second. Making the conversion to miles per hour, we find that, on average, the car is traveling (drumroll, please)...

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4.09 miles per hour.

Yeah, I know. It's a little underwhelming. I thought it would be faster! If your arms are longer and you're able to get the same number of "laps" (or more), it'll be a little bit faster, but not by much. 

But, we're not quite done. We've been looking at how fast a two-inch car can travel 26 inches; what if we scaled up the car and the distance up to real-world values, but kept the time the same? Or, in other words: if there were a little tiny man sitting in the cars we were driving back and forth, how fast would he think he was moving?

To find out, we multiply the distance of 26 inches by the factor that each car is scaled down by. It's different for each car; let's look at the BMW, which is at 1:57 scale. We multiply 26 inches by 57; that's 1484.78 inches. As before, we divide that by the BMW's time, which was 0.337 seconds. That's 4404.837 inches per second, or, in other words...

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250.275 miles per hour!

The good folks at Quora tell me that a single engine plane like a Cessna flies at a speed somewhere around 130 miles per hour. Therefore, since your toy car is moving at nearly twice that speed (from it's perspective), it is completely logical and rational for you to pick it up off the ground and make it fly. We want to be scientifically rigorous, after all.

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Thanks for the question! Have fun with your kids!

-Frère Rubik

P.S. For kicks, here's my scratch paper for this question, which contains all of the individual car times/speeds. If you're having trouble reading it and want to know the specific numbers, feel free to shoot me an email.

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