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I started wondering today, how does one calculate the moving velocity of a vehicle if they change their tires?

I know there are a lot of companies that make the gauges for tuners that change wheels, but what is the math involved in this? If for instance I wanted to manually set my speedometer or tune the speedometer myself, what exactly what you have to take into account and how would you calculate everything correctly?

I didn't find this appropriate for the mathematics SE, so I posted it here.

thanks for the help.

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The only thing your car can sense is the number of turns of some part of the drive train. In the old days it was a gear right off the transmission, and today it's an electronic sensor in the transmission or somewhere in the drive train near the wheels. It's going to be a sensor that can determine how quickly the wheels are rotating.

The way to calculate speed is the distance traveled divided by the time it took to travel that distance.

Table of calculating speed as a function of time and distance

The only thing that changes when you change the tire size is the distance traveled. So, for every revolution of the tire, you travel a distance equal to the circumference of the circle that is the tread of the tire. (Assuming the tire is not slipping).

The circumference (C) of the tire is calculated by C=2πr (2 times pi times the radius of the tire). So the distance from the center of the wheel to the outside edge of the tire, times 2, times pi (3.14159). Do this calculation for both the old and the new tire.

To get the percentage change in the speedometer, take the difference between the old and new distance, divide by the original distance, and multiply the result by 100. This will give you the percentage speed change from one tire to the next.

I hope that helps!

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    For the purposes of a tire, the radius is measured from the center of the wheel to the ground or floor. Multiplied by 2pi that is the rolling circumference of the tire, not as measured from the center to some other outside edge of the tire. – BillDOe May 24 '16 at 22:51
  • Interesting, I hadn't considered that. Since the sidewall is slightly flatter at that point from holding up the weight, your quite right. Good catch! – cdunn May 25 '16 at 2:32
  • Thank you for the excellent answer. This is why SE Exists! – cloudnyn3 May 26 '16 at 3:50
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To calculate the actual speed you need:
Transmission gear ratios
Final drive ratio
Tire diameter
Engine RPM

[tire diameter (in inches)] * [pi] * [1/(gear ratio * final drive ratio)] / [in/ft] / [ft/mi] * [RPM] * [min/hr] = MPH

Constants: pi = 3.14159
in/ft = 12
ft/mi = 5280
min/hr = 60

I'm sure I could simplify this equation, but this is good for seeing all the necessary parts and conversions.

So I watch the tach and speedo to determine MPH per 1000 RPM in higher gears (3rd,4th,5th). On my car its about 11, 15, 19.5.

calculated:
tire diameter - used an online calculator - 195/55R15 = 23.4in
gear ratio (3, 4, 5)- 1.360, 1.034, 0.787
FD ratio - 4.40
RPM - 1000

3rd: 23.4 * 3.14 * ( 1 / ( 1.360 * 4.40 ) ) / 12 / 5280 * 1000 * 60 = 11.6
4th: 23.4 * 3.14 * ( 1 / ( 1.034 * 4.40 ) ) / 12 / 5280 * 1000 * 60 = 15.29
5th: 23.4 * 3.14 * ( 1 / ( 0.787 * 4.40 ) ) / 12 / 5280 * 1000 * 60 = 20.1

This seems to be about correct. There are a couple things that will throw this off such as tire diameter not being exactly as advertised, tire wear, and the tach and speedo not being 100% accurate. Doing this as one equation and using a calculator's pi gives more accurate results due to significant digits.

You can also use a GPS (or a speedometer app on your phone) to view your speed.

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