The simplest way to put it goes like this:
The different gears in a car correspond to different *gear ratios* — how many times the engine crankshaft turns for every time the drive shaft — the part connected to the wheels — turns.
Engines operate best when the engine speed — measured by the *tachometer* — is in a certain band for each given gear set.
Pushing in the clutch physically disconnects the crankshaft gear from the drive gears, and then you can move the gear selector, which causes a different drive gear to lock up against the crankshaft.
By doing this, the driver can ensure that the car is operating in the right ranges for optimal handling.
So there are different gears in the gearbox, these combine to have different ratios of power. Like converting 10 engine rotations, into 1 turn of the driveshaft/wheels.
The clutch disconnects the drivetrain from the engine so that the gears can move into place.
https://www.reddit.com/r/explainlikeimfive/comments/oqrtvy/eli5_how_does_a_manual_transmission_work/ here is another thread on it.
A manual gearbox have two main shafts, an input and an output shaft. There are cogs on these shafts of different sizes which can slide on the shaft but rotate with it. The gear selector is mechanically moving these cogs so that on set of cogs mesh. This locks the ratio of the input and output shaft to a fixed value called the gear ratio.
When you change gear you first have to remove forces and mass from the system. So you disengage the clutch to disconnect the engine from the input shaft. Then you put the gear selector in neutral. The input shaft is now rotating completely on its own innertia and is very light weight. So when you select the next gear and the cogs of the new gear start to mesh a set of syncronizing rings on the cogs will mesh first to change the speed of the input shaft such that the cogs will mesh perfectly. Once you put the car in gear the cogs mesh fully, you can often feel it click into place from the initial syncronizing ring engagement to the cogs, the input and output shafts are in this locked ratio. Then you start engage the clutch to connect the engine to the input shaft again. The clutch is allowed to slip a bit when it is half engaged so that the speed of the engine and the input shaft on the gearbox is allowed to syncronize.
Specifically about race cars the syncronizing rings and the clutch are technically not needed to change gear in a car. But getting the gears to mesh without these is practically impossible for a novice driver and even hard for most experienced drivers. However using the sycronizing rings and clutch does wear them out and lose you power. For a race car you also have the issue that the engine can be too powerfull for the clutch and if abused it can ruin the clutch. For this reason racing drivers tend to learn how to match the engine speed and car speed with the gear ratio of the gear they switch to and make sure to have as little throttle application as possible when selecting the gear and using the clutch so the forces through the clutch and syncronizing rings are as low as possible. There are even real examples of race cars that have had clutch failures during a race and the driver still finished at a good pace and even some rare cases of clutches not being installed into some race cars to improve performance.
I actually happened across [this video](https://youtu.be/9UmrCl2nLKM) earlier today, it’s mostly about the clutch, but it also has a quick section on the transmission (mostly focused on the reverse gear) with an open example. Not really an explanation, but I figure it’s a good visual example.
Be warned, the opening theme is very loud and annoying.
#ELI5
Have you ever ridden a [bicycle with multiple gears](https://cyclejourneys.co.nz/wp-content/uploads/2020/02/Drivetrain.jpg)?
The rest of this answer is based on the assumption that you have had that experience.
So you already know that “however you set the gears” makes it easier to pedal up hills, or easier to go faster, basically you pick the gear that works best for your current situation.
If you ever took a closer look at the bike, you see there are actual gears ([rings with teeth on them](https://m.media-amazon.com/images/I/51DsqnbrI6S._AC_.jpg)). In fact, that bicycle has several different sized rings in the back (at the wheels), and maybe another set of different sized gears in the front (at the pedals).
When you manually change gears, you’re selecting a different ring for the chain to go around.
Inside a car it’s kind of the same thing. The engine is “pedaling”, and there are gears with different numbers of teeth. Which gear you select means a different number of teeth are engaged, and it makes it easier to use the same force (like pedaling) to go up a hill slowly, or across a flat road more quickly.
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