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.

Unit circle with point P(cos t, sin t) marked. Tangent to circle at P drawn and labelled v

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.

  • 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 \(\sin t\) and \(\cos t\)?

Here are some questions to ask yourself when looking at the diagrams. (The diagrams with questions are also provided on cards.)

What is the magnitude and direction of \(\mathbf{v}\)?

Why is \(P\) in this position at time \(\quantity{t}{s}\)?

How are vectors \(\mathbf{v}\) and \(\overrightarrow{OQ}\) related?

How are the two triangles related?

Why is \(\mathbf{v}\) given by the vector shown?

Why are these the coordinates of \(Q\)?