## Problem

Particle $P$ is moving anti-clockwise round a circle of radius $\quantity{1}{m}$ at speed $\quantity{1}{m\,s^{-1}}$.

The particle is at point $A$ when $t=0$ and its position at time $\quantity{t}{s}$ is shown in the diagram below.

Combine some of the following diagrams to explain why the velocity vector $\mathbf{v}$ is $(-\sin t, \cos t)$. You may be able to do this in more than one way. (The diagrams are also provided on cards.)

• How have you used the fact that $P$ is moving at unit speed on a unit circle?

• What does this tell you about the derivatives of $\cos t$ and $\sin t$?

Velocity is the rate of change of displacement, so its components are given by the rates at which its $x$ and $y$ coordinates are changing with respect to time. Therefore $\mathbf{v}=\left( \dfrac{dx}{dt}, \dfrac{dy}{dt}\right).$