kinematics - Position in circular motion

Is it possible to determine the position of a particle undergoing circular motion, in x-y coordinates, at any given time and velocity?

I'm thinking it has something to do with $\pi$

Answer

Yes, unless you want to be very picky. The coordinates are described by the trig functions sine and cosine.

Suppose something is moving on a circle with radius $R$, so its $x$ and $y$ coordinates obey

$$x^2 + y^2 = R^2$$

If it moves at a uniform velocity of $v$, then the angle its path subtends is linear in time (by dimensional analysis), so we can define the angle $\theta$ as

$$\theta = vt/R$$

One full rotation turns out to be an angle of about 6.28, but it's really a transcendental number, $2\pi$.

If you draw a circle, then draw the angle from the equation above, you find a unique intersection point that tells you where the object is. If you project that point onto the $x$ and $y$ axes, you've graphically determined the coordinates.

The coordinate you obtain, for a given angle, is linear in $R$ (again by dimensional analysis, coupled with a feature of Euclidean geometry - that it has no innate length scale), so if we divide the coordinates by $R$ we get two functions that map $[0,2\pi) \to [-1,1]$ because $-1$ and $1$ are the minimum and maximum values, since $-R$ and $R$ are the minimum and maximum values of the coordinates.

These functions are give the names "cosine" and "sine", and we say

$$x/R = \cos\theta$$

$$y/R = \sin\theta$$

The sine and cosine functions can be represented as infinite series,

$$\cos\theta = 1 - \theta^2/2 + \theta^4/24 - \ldots$$

$$\sin\theta = \theta - \theta^3/6 + \theta^5/120 - \ldots$$

We know many of the properties of these functions, and they are used extensively in mathematics and physics. You can calculate their values to high precision with a computer. However, we cannot simply write down exactly what $\sin(1)$ is in any simple way that you can explain to an elementary-school student. If you want to know what it is, you can simply use a calculator, like this.