because if there is a dry tire and there is rain the tire would slip the driver would probably die in a crash.But if you have a wet tire on a wet course there a 75% chans that you will not slip
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Tyres are need to make good contact with the road to keep the car form slipping (called traction). Different road conditions need different tyre compositions and tread patterns to attain maximum grip, so in racing different tyres are used depending on the conditions.
The speed of an f1 car depends on many things. It's engine capacity, gear shift ratio, aerodynamic efficiency of the car body etc. Since different f1 cars are made by different teams, they don't essentially have the same features. So cars from one manufacturer is faster than the other.
Driver talent is also equally important. Two drivers, given the same car may drive at different speeds
Now a days it is controlled by regulations, but the general shape was designed by engineers to get the maximum down force with the least amount of drag and still have enough cooling for the radiators, brakes, etc.
Rear wheels are actually wider than the front ones. This is due to the amount of force and stress put onto the tyre in the name of torque as well being able to cope at going speeds up to 200mph+ without them splitting or falling apart like standard family car tyre may do. So the wider tyres gives more contact patch onto the road giving a greater surface area resulting in less load required at any one point of a tyre. Same for road cars but sizes very depending on where and how they're/would be used.