Another engineer's opinion...
Thanks, FD, and I hope GC didn't think I was being abrupt; I had to sneak in a post because We are Cleaning the Basement and that's all I could fit in... savvy? :wink1: (as Captain Jack Sparrow would say).The difference in the case of speed and velocity emerges really clearly when you look at circular motion of some mass at a constant speed. Is the velocity constant too? No, because the velocity has two components, one parallel to the tangent of the motion and one perpendicular to the motion, the normal component. The tangential component is constant, but the normal component isn't (hence, the time derivative of the normal component of velocity doesn't vanish). So a mass orbiting in a circle with constant speed has a non-null acceleration, which is why torque exists even when a mass is rotating at constant speed.
There's one other thing: I don't think this stuff is reducible, if that's the word, to basic mathematics. The whole of classical Newtonian mechanics rests on the central observation that momentum, the vector quantity m
v, usually written
p, is conserved in any closed interaction. That is, the time derivative of
p is zero. So that means that where momentum is not conserved, the system is not closed, and the intervention from outside the system is characterized as a
force. Since the measure of the force is the degree to which the time derivative of momentum differs from zero, the conservation of momentum entails that
F = d
p/dt, which is just another way of writing Newton's `second law', which isn't really a law, but rather a claim about a particular physical quantity—i.e., that it doesn't change over time in an isolated system. But that needn't be the case. It's a fact about the universe, not about mathematics. The fact that both linear and angular momentum are conserved quantities, among others conserved quantities, corresponds to constraints upon the universe we happen to live in. It's not entailed either by the foundations—or any particular elaborations of the foundations—of mathematics. Classical physics relies on mathematical relationships, but its central premise, the conservation of momentum, is just a fact about how things are....