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DanArmak comments on Harry Potter and the Methods of Rationality discussion thread, part 5 - Less Wrong
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It seems that they might act in a hybrid Aristotelian/Newtonian manner. Certainly in canon they talk about broomsticks having maximum speed not maximum acceleration. And people have trouble pulling from being near to hitting the ground, something which sort of makes sense in an Aristotelian framework because objects want to go to the ground.
Outside canon, the movement of the broomsticks in the movies does seem to be a definite mix but this is likely more due to standard movie physics than anything else.
It makes sense by postulating that a broomstick always goes where it's pointed (no Newtonian momentum), but there is a maximum angular speed for turning the broomstick. The rider applies force to turn the broomstick, which means there's resistance, so it's not difficult to assume that the resistance creates an effective maximum angular speed.
This doesn't sum up to Newton, of course, because this maximum angular speed isn't dependent on current linear speed.