# Speed to RPMS and Back Again: A Simple Tutorial

Discussion in 'General Questions' started by roughrider, Feb 28, 2013.

1. ### roughriderMember

Hi all,

I've been trying to wrap my mind around gear ratio calculations so I could convert rpms to distance, or distance to rpms quickly and easily, and, preferably, in my head as I'm riding.

Well I figured out a simple method that I'd thought I'd share. Since I'm in the U.S., I'm used to feet, inches, and miles, so I'm using those here, but the method works with any unit of measurement; in fact, metric is WAY easier.

First, I'm working with an HT engine that has a 4.1:1 gear reduction.* I also have a 48 tooth rear sprocket and a ten tooth drive gear. So that gives me a further 4.8:1 reduction. Make sense so far?

If you had, say, a 36 tooth rear, you'd have a 3.6:1 reduction. You just divide the number of teeth on the smaller gear into the number of teeth on the bigger gear to get that number.

To get your total, you multiply the two ratios. So for my bike, 4.1 • 4.8 = 19.68

This is the first of the "magic numbers." We'll call it "A." That's my happy way of describing a mathematical "constant." We could also call it "Thomas Jefferson" or "Fluffy McScatosbreath," but you'd get tired of having to write that out over and over.**

If you had a 36 tooth rear sprocket, all else being equal, your magic number "A" would be 4.1 • 3.6 = 14.76

Save that number. You'll need it.

Next, you need to figure out how far your drive wheel travels in a single revolution.

To do this, you need your actual tire diameter. I was surprised to find that the big, fat rear tire I have actually gives me 27 inch diameter. That's about the same as a "700C" like a lot of modern road bikes and mountain bikes have. I suggest you take an actual measurement. This needs to be EXACT.

Of course, to get the circumerence of a circle, you multiply the diameter times "pi" (π). This is approximately 3.14, but I suggest you use a calculator that has π built-in.

My 27 inch tire thus has a circumference of approximately 84.823 inches. For precise calculations, I use the number stored in the calculator. For doing math in my head, I just use 85 inches. It so happens that my circumference is very, very close to seven feet. That was lucky.

Again, A = 19.68 ≈ 20 ("≈" means "approximately equal.")

Thus, with my setup, for every 20 revolutions of the engine, the bike travels very close to seven feet. If I have a 36 tooth sprocket, the engine would have spun about fourteen and three-quarter revolutions. Are you still with me?

We are almost there.

From these numbers, we can easily convert the number of revolutions to feet travelled. At 1,000 rpm, my rear tire turns about 50 times. (1,000 / 20 = 50) Or, to be perfectly precise, rpm/A = wheel revolutions. With a 36 tooth sprocket, that would be 1000 divided by 14.68, which is about 68.

Now, if we take our wheel revolutions and multiply that by the tire circumference, we get the distance travelled in one minute. So in my case, that's fifty times seven, which equals 350 feet. With a thirty-six tooth sprocket, that would be 68 • 7 = 476 feet.

Purposely, I've kept the calculations in minutes and feet. It just keeps the numbers nicer. But since we want to toggle between rpms and mph, we need to do some converting. Feet per minute to miles per hour takes two steps. You can do it in any order, but I'll just multiply feet times 60 to get feet per hour; thus, 350 • 60 = 21,000 feet per hour. Dividing that by the number of feet in a mile (5,280) gives us 3.97727.. ≈ 4mph.

OK! So when my engine is turning 1,000 rpms, I'm going 4mph. Cool!

Here's the fun part: Divide the revolutions per minute by the miles per hour and you get a new constant. This is our second Magic Number. In this case, it is 250. Ah hah!

Now, I want you to make a little jump. Let's say I'm going 20 miles per hour. If I multiply 20 times 250, I get 5,000. That's my engine rpm! If I know my engine rpm, say, 4,000, I divide 250 into 4,000 and I get my speed--16mph!

See how this works?

Using our thirty-six tooth sprocket example, we get different results. We get a different Magic Number. With the 36 tooth sprocket, in one minute, at 1,000 rpms, the bike will have travelled 476 feet. In one hour, that's 28,560 feet, which is about 5.4 miles. Dividing 1,000 by 5.4 gives us very close to 185. That's the Magic Number for the 36T sprocket.

So let's see... If my engine is doing a screaming 7,500 rpms, dividing that by our constant gives us just over 40mph!

Of course, in the real world, there are other variables, like the friction loss of the tire spinning a bit faster than the actual speed travelled due to the ever mounting resistance you encounter as you go faster and faster. Those problems can be solved after about one semester of college calculus, but we do NOT want to go there, do we? Rather, you can use this theoretical, approximate, data to compare with your speedo or GPS data. You'll find that at lower speeds, it's pretty close, but the faster you go, the more error. In fact, you could use your collected data (as I have) to plot a curve that could be used to derive the function for the advanced math.

But let's not. Yet, for math afficiandos, it's similar to "the cannonball problem."

Here's the challenge. What's YOUR magic number?

If my explanation has not made sense, I'll be happy to step you through this process... at least for the next few weeks as I monitor this thread.

And btw, I have purposely limited this example to a single speed motorized bicycle. You guys with shift kits have probably already done a bunch of careful calculations. Even so, the same process works. You just have more calculations to get to a SET of magic numbers.

*4.1? How'd you get that? Well, you count teeth. You count the teeth on the little gear and count the teeth on the big gear, and you divide the smaller number into the bigger number. (Or, you do what I REALLY did, which was just read up on it.) But, basically, the little gear is turning 4.1 times for each revolution of the big gear.

**There is a tradition among mathematicians to use letters at the beginning of the alphabet for constants and numbers at the end of the alphabet for variables. A further tradition is to use Capital Letters for numbers which are themselves the result of calculations. It is just a tradition, but it can be helpful.

Last edited: Feb 28, 2013

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2. ### grinningremlinActive Member

H.o.l.y. C.r.a.p.!You're too smart for me.Very concise, especially from my "eyeballers" point of view.I can "do" some math (musician/build guitars) but couldn't explain any of it, numbers are greek to me.
Well written RR.

3. ### roughriderMember

Thanks, but, well, anyone who can do music can do math. It's the most mathematical of the arts. But truly, GG. I'm trying to be helpful. So if you can give me the gear specs on your bike, I could chug the numbers for you. That might help others too.

Last edited: Feb 28, 2013
4. ### darwinWell-Known Member

Personally I like the KISS principle, sorta like 60mph = 1 mile per minute etc.......30mph = 1/2 mile per min.

5. ### roughriderMember

Heh. I dunce thought I WAS keeping it simple!

Still, you just need your output shaft reduction, your drive gear ratio, and your rear wheel diameter for ALL the calcs.