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First, the electron is in a probability cloud about the nucleus of an atom. So it's not really whizzing anywhere.
Second, It doesn't respond to Newton's equations in the same way.
An electron must remain between specific energy states within an atom and so it doesn't respond to things like friction the way large bodies do.
Electrons do however respond to certain frequencies of light by moving up an energy state and give off the same light when they move down. This is where the light from a hot piece of metal comes from as well as the light from the aurora.
can it not be said to be "whizzing" round in as much as it generates a magnetic field which I take it is why Hydrogen atoms like to go round in pairs.