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If you drive a stick-shift car, then you may have several questions floating in your head.

How does the funny "H" pattern that I am moving this shift knob through have any relation to the gears inside the transmission? What is moving inside the transmission when I move the shifter?
When I mess up and hear that horrible grinding sound, what is actually grinding? What would happen if I were to accidentally shift into reverse while I am speeding down the freeway? Would the entire transmission explode?
 

Cars need transmissions because of the physics of the gasoline engine. First, any engine has a redline -- a maximum rpm value above which the engine cannot go without exploding. Second, if you have read How Horsepower Works, then you know that engines have narrow rpm ranges where horsepower and torque are at their maximum. For example, an engine might produce its maximum horsepower at 5,500 rpm. The transmission allows the gear ratio between the engine and the drive wheels to change as the car speeds up and slows down. You shift gears so the engine can stay below the redline and near the rpm band of its best performance.

Mercedes-Benz Actros, manual transmission

Ideally, the transmission would be so flexible in its ratios that the engine could always run at its single, best-performance rpm value. That is the idea behind the continuously variable transmission (CVT).

A CVT has a nearly infinite range of gear ratios. In the past, CVTs could not compete with four-speed and five-speed transmissions in terms of cost, size and reliability, so you didn't see them in production automobiles. These days, improvements in design have made CVTs more common. The Toyota Prius is a hybrid car that uses a CVT.
 

The transmission is connected to the engine through the clutch. The input shaft of the transmission therefore turns at the same rpm as the engine. A five-speed transmission applies one of five different gear ratios to the input shaft to produce a different rpm value at the output shaft. Here are some typical gear ratios:

The five-speed manual transmission is fairly standard on cars today. Internally, it looks something like this:

Keep in mind that the shift lever has a rotation point in the middle. When you push the knob forward to engage first gear, you are actually pulling the rod and fork for first gear back.

You can see that as you move the shifter left and right you are engaging different forks (and therefore different collars). Moving the knob forward and backward moves the collar to engage one of the gears.
 

Reverse gear is handled by a small idler gear (purple). At all times, the blue reverse gear in this diagram is turning in a direction opposite to all of the other blue gears. Therefore, it would be impossible to throw the transmission into reverse while the car is moving forward -- the dog teeth would never engage. However, they will make a lot of noise!

Synchronizers
Manual transmissions in modern passenger cars use synchronizers to eliminate the need for double-clutching. A synchro's purpose is to allow the collar and the gear to make frictional contact before the dog teeth make contact. This lets the collar and the gear synchronize their speeds before the teeth need to engage, like this:

The cone on the blue gear fits into the cone-shaped area in the collar, and friction between the cone and the collar synchronize the collar and the gear. The outer portion of the collar then slides so that the dog teeth can engage the gear.

Every manufacturer implements transmissions and synchros in different ways, but this is the general idea.

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