So lets look at this in a basic sense first. Assuming 2 cars are the same weight, length, gearbox, tire sizes, etc… yet different power from the engine. The car with more power will be faster.
In the real world however, two cars are not always the same. Things like different rev limits, gearing, tire sizes, and even CG position and driving wheels can change how fast a car accelerates. Take something like a Honda S2000. Its engine is designed to rev up to 9,000 rpm. But another car, such as a Mustang GT may only have a 6,000 rpm redline. Both these cars will have to have a different set of gears to achieve a balance between those powers, which will play into the acceleration difference between them if all variable of torque and power from the engine are otherwise the same. Some manufactures want their first gear to top out before 60mph which makes it so you have to shift, thus have time where there is not power pushing you forward as well. You can change some of these variables, like switching from a 4.1 diff to a 4.4, which does change and improve or degrade performance for cars in certain areas as well.
Still, in the most general sense, a car with more power will almost always have a faster 0-60 time.
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