I’m going to use made up numbers to keep the math easy.
Let’s say you have a gasoline engine whose slowest operating speed is 700 rpm (revolutions per minute – how many times the crankshaft spins each minute) and the fastest is 7000 rpm, which means the fastest speed is **10x** the slowest speed.
If you put a wheel about 28 inches in diameter on the end of that spinning shaft, the lowest speed is 5 mph, and 10x that is 50 mph top speed. But there’s a few problems with this. The nature of a gasoline engine is that it generally creates more power the faster it goes. The problem in this case is that you need more power at slower speeds so you can accelerate up to cruising speed where you need the least power. Another problem is that the engine wears out proportionally to the **square** of engine speed, so speeding it up 10x results in **100x** the wear. Not to mention that we need to go slower than 5 on occasion, and faster than 50.
This is what gears are for. Gears are effectively _multipliers_ of the engine rpm. You can multiply by numbers smaller than one to make it smaller, and higher than one to make it bigger. The auto engineer looks at the characteristics of the engine and determines the best rpm for cruising. Let’s say that our engine should do 75 mph but at only 3500 rpm (half the top speed). If the gear triples the top speed this would work. This becomes our top gear.
But now we definitely won’t have enough power to speed up to 75. So the engineer looks at the power the engine makes at each rpm and figures out what the ratio between the lowest and highest usable rpm’s are. Then you divide our final 1/3 ratio (one input rpm to three output rpm, 1/3 as a fraction) by this new calculated ratio to figure out what your second top-most gear should be. If this new gear is calculated to be a good fit for low-speed driving then you stop adding more gears. If it’s not a good low speed gear, repeat the calculations and keep adding gears until it works in a driveway or parking lot.
Because this depends on the characteristics of the engine, and technology changes all the time, the number of gears have changed over time. Most cars in the seventies had 3 gears. By the nineties it was 4 or 5 for regular cars, 6 for performance cars. Now, 7-10 gears is common in automatic transmissions.
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