Is velocity a function of displacemnt?

Question

The velocity $\displaystyle\vec{v}$ of a particle $=\frac{d\vec{x}}{dt}$. So surely this means that $\vec{v}$ is dependent on the position of the particle?

Answer 1

Consider a particle which passes through a particular point going in one direction, then loops around and passes through the same point again going in the opposite direction. Then at some particular position, the particle has two different velocities. So the velocity of this particle is not a function of its position, in the sense that if $v(t)$ denotes the velocity at time $t$ and $x(t)$ denotes the position, there is no function $f$ such that

$$v(t) = f(x(t)).$$

What the relationship $v = \frac{dx}{dt}$ tells us is that the velocity as a function (rather than the velocity at any particular time) is a function of the position as a function (rather than the position at any particular time). Intuitively, to know the velocity at a particular time $t$ it doesn't suffice to just know the position at time $t$; you need to know the position in some small open interval $(t - \varepsilon, t + \varepsilon)$ around $t$.

Answer 2

In mathematical terms, it sounds as if you're asserting:

"The first derivative $f'(x)$ of a function $= dy/dx$. So surely this means that $f'(x)$ is dependent on the value $y = f(x)$."

A moment's thought should reveal that the derivative of a function and the value of a function are completely independent (so long as you're speaking of functions in general, not, e.g., solutions of a differential equation).