Arnold said:
if you accellerate quickly, and accellerate slowly (up to the same rpm). does it use the same amount of fuel?
This one isn't easy to answer.
Conventional wisdom states that driving gently you get better MPG. Totally true. That isn't what the question asks though.
A theoretical petrol engine is most fuel efficient (amount of fuel burnt to power produced) when the throttle is wide open. This is because pumping loses are reduced. This is true across the rev range.
You can otherwise use a large amount of power to accelerate quickly, or a small amount of power to accelerate slowly. The change in kinetic energy is the same, so the amount of energy the engine produces is the same whatever. If 100bhp takes 5s, 50bhp will take 10s.
The amount of energy an engine makes is related to the amount of fuel it burns, and then you take into account efficiency. Both types of acceleration use the same amount of fuel before you take throttle position into account. But the quicker acceleration has a wider throttle opening, so is working more efficiently.
Hence, accelerating quickly uses less fuel during that period of acceleration. This ISN'T MPG. You will have travelled less distance for the same amount of fuel - but that wasn't the original question.
In the real world, in actual driving, it is different. Hard acceleration will generally cause greater fuel enrichment. Hard acceleration is normally accompanied by changing gear at higher rpm. Hard acceleration normally means stop-start driving (constant speed uses less fuel). Hard acceleration means wheel spin and slipping clutch.