How the formula was derived

The formula is based on first determining the speed of the wheels in revolutions per minute. This is accomplished by starting with the engine speed and working through the drivetrain until one gets to the wheels. The best way to show this is by example. 

Let's say that a certain vehicle is traveling in second gear with an engine RPM of 2000 revolutions per minute. Now, in this particular car, the transmission ratio in second gear is 2:1. That is to say, for each 2 revolutions of the engine, the output shaft of the transmission will rotate once. So, the output of the transmission, and thus the drive shaft will be rotating at 1000 RPM. In the rear axle of this vehicle (assuming it is rear wheel drive), a further gear reduction is encountered in the ring and pinion gears. For the sake of simplicity, let's assume that the ratio of this gear set is 4:1. In other words, for each 4 revolutions of the drive shaft, the axle shafts which connect to the wheels revolve one time. So, in our case, the 1000 RPM speed of the drive shaft is reduced to 250 RPM for the axle shafts. Since the wheels are bolted directly to the axle shafts, they will also rotate at 250 RPM.

Now comes the tricky part. We have to determine how far a wheel travels per revolution. This can be determined by the diameter of the tire. The diameter is determined by measuring along the side of the tire, making sure that the ruler crosses the center of the wheel as shown:

Once the diameter is determined, the distance per revolution can be calculated. The distance travelled per revolution is equal to the circumference of the tire, so all one needs to do is determine a tire's circumference to determine how far it travels. To do this, you simply multiply the diameter of the tire by the mathematical constant Pi (3.1416). So, if our vehicle's tires are 25" in diameter, the distance travelled per revolution is equal to 25*3.1416 or 78.5 inches per revolution. Recall that the axle speed on this vehicle was 250 RPM, we can see that the tire will cover 78.5*250 or 19,625 inches per minute. Of course, we're looking for miles per hour, so this figure will have to be converted.

To convert from inches per minute to miles per hour, we can first determine how many inches are in a mile, and then determine how many minutes are in an hour. It's pretty obvious that there are 60 minutes in an hour, so that won't have to be calculated. Considering that there are 5280 feet per mile and 12 inches per foot, it can be seen that there are 63,360 inches in a mile. Back to our vehicle, we had a speed of 19,625 inches per minute. To convert to miles per minute, we divide 19,625 by 63,360 to get 0.310 miles per minute. Since we need miles per hour and there are 60 minutes in an hour, we simply multiply 0.310 by 60 to get 18.6 MPH. 

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