Tittle says it all how does a mechanic modify the engine/car components to make it faster.
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Engines work on explosions. You need to either make the explosions bigger, or make them happen faster (more often) to make more power to go faster.
They make explosions bigger in a few ways, such as increasing cylinder size or by forcing more air and fuel into the cylinders. This is typically where power is added.
Making the explosions faster is a little harder because it is a matter of precision and balance so the engine doesn’t just blow itself up or vibrate apart. Faster is also meaningless if the explosions aren’t big enough to create enough power to move the car.
More power is the most straightforward. Power comes from burning fuel. This generally means getting more air into the engine so that more fuel can be pushed into the engine(more fuel without more air=all the fuel won’t be burned and is just wasted), or ensuring that the fuel burns more efficiently(e.g. higher compression or better fuel injectors), or reducing friction or other losses (bigger exhaust for example).
Or if “faster” means faster on a track, then cornering and acceleration becomes important. Better cornering helps because you don’t need to slow down as much. Better tires provide better grip. Better suspension keeps tires in contact with the road to maximize the use of the grip. Lower weight helps accelerate faster, take corners faster, and brake faster.
Engines make power by mixing air and fuel then igniting it. The more air and fuel you can ignite the more power you make. So modifications are done to force more air into the engine then more fuel is added to compensate for more air. Bigger bang. Mo powa baby.
To answer your question about “how does a mechanic modify the engine/car components to make it faster”, it helps if you understand how an internal combustion engine (ICE) makes power, which others have already mentioned. Extracting more power from a “slow” engine generally involves removing restrictions and inefficiencies in the entire system of the vehicle, not just engine
Removing weight is less mass the engine has to move. Increasing airflow into the intake makes stronger explosions without directly modifying the engine (turbo- or supercharging the engine is an extreme version of this called “forced induction”, versus the term “naturally aspirated” which is how most stock cars operate). Changing the fuel system (even completely changing the type of gas the car takes) creates even stronger explosions. Removing exhaust components reduces a non-negligible amount of restriction on gases leaving the engine. Bigger tires transmits more power to the asphalt. Changing gear ratios can give you more acceleration but reduced top end. Even changing the vehicle’s body parts can increase aerodynamic downforce, providing additional gains to traction depending on the car (i.e. putting a spoiler on your Civic? Just for show. Putting a spoiler, canards, and diffuser on an F1 car? You create enough downforce that the car can almost literally drive upside-down).
All of these incremental modifications can add up to big gains in the overall “speed” of the car, but it really, really helps to start with an already performance-oriented motor. You also can have $12,000 motorcycles that are faster than $120,000 sports cars, almost purely because of the horsepower-to-weight ratio.
Generally, cars use internal combustion. This involves mixing air and fuel, squeezing it, and making it explode. The more force the explosion makes, the more power. In theory, this means the more air/fuel mixture you can cram into each cycle, the more powerful the explosion. To add power to an existing engine, generally you can add forced induction. Turbos or superchargers are most common. This compresses the air to allow higher density, allowing the addition of more fuel.