Auto Math 101
   
Several people have asked me lately how to calculate speeds in gears or engine rpm at a given speed. These calculations aren't complex but they do involve a few steps.

The first step is to gather some information. You will need (1) your final drive ratio, also known as the rear end ratio; (2) the ratios of your transmission gears; and (3) the effective radius of the driven tire. The first two chunks of data you can usually get from any service manual. The third one, effective radius, you will need to measure because it varies from one size of tire to another and even from one brand of tire to another in the same size. Simply measure from the ground up to the exact center of the wheel. Your tape measure will have it in fractions, so you will need to convert the fraction to a decimal equivalent. No big deal.

Now the calculations begin. You will need to figure out the "revs per mile" of your tire. Do this by doubling the effective radius and multiplying that number by 3.1416 (You can use 3.14 if you are doing this on stone tablets with a chisel because your calculator's battery died. The difference will be insignificant.). The resulting figure is the circumference of the tire, which is also the number of inches your car travels for one complete revolution of the tire. Since there are 63,360 inches in one mile, dividing 63,360 by the tire circumference will give the "revs per mile" figure. Hang on to this number because you will be using it a lot.

The significance of this number is that if your tire is turning that many revolutions in one minute (rpm's), you are going 60 mph. Let's use my Sentra SE-R as an example. The effective radius is 10.75 inches, so:

  10.75 x 2 = 21.5
  21.5 x 3.1416 = 67.54
  63,360 / 67.54 = 938

Now all you have to do is figure out how fast the tire is turning. This is where you need those gear ratios. Calculate the overall ratio for the gear you are interested in, or do them all. Multiply the transmission ratio by the final drive ratio. I'll just do 5th gear in the Sentra. Final drive ratio is 4.18 and 5th gear is 0.76, so

  4.18 x 0.76 = 3.18

If I want to know how many rpm the engine will be turning at 60 mph in 5th, I multiply the tire revs per mile by the overall ratio:

  938 x 3.18 = 2983

(or approximately 3000 rpm)

But suppose you only know your engine rpm and you want to figure out how fast you were going in a particular gear. Say your speedo is broken (must be a roadster) but you know you were cooking down this road the other night at 6,000 rpm in 3rd gear. How fast was that?

Easy to do. Divide your engine rpm by the overall ratio in that gear. This gives you the tire rpm. Using the Sentra again, the overall ratio in 3rd gear is 5.39, so

  6,000 / 5.39 = 1,113 tire rpm

Since 938 rpm would be 60 mph, you can see this is cooking right along! To figure out how much, divide the tire rpm by the revs per mile and multiply the result by 60. This will be the speed in miles per hour.

  1,113 / 938 = 1.19
  1.19 x 60 = 71.4 mph

If all this was put into the form of a formula, it would be

  (engine rpm / overall ratio)    
  ------------------------------------- x 60 = mph
  (63,360 / (radius x 2 x 3.1416))    

which certainly looks more impressive, but is more difficult for most people to use. What we did was start on the bottom right and work our way left, then up, then right to get the result.

Now that wasn't so hard after all, was it?