Suppose a centipede is bicycling around a circular track at a constant speed. We want to be able to describe the position, velocity and acceleration of this noble creature as a function of time. We'll call the radius of the track R and the the speed of the centipede v.
The angle 
varies linearly with time so we can say that
where 
is a constant. It's often
referred to as the angular velocity since its the rate an
angle changes with time. In the present example it is just
some constant with dimensions of inverse time.
So now we can describe the position of the centipede in terms of it's x and
y components as seen in green in the figure. A little trigonometry
gives the 
and 
. So substituting
and expressing the result in terms of vectors:
Good, now we have a well defined mathematical expression for the position of the centipede as a function of time. We can now differentiate it once to obtain the velocity
So what does this say about the speed? We know how to take the magnitude of a vector right? Remember Pythagaros? (Did you know he worshiped beans?) So we have
So 
. So that tells us how to relate this mysterious
to v. Now we can differentiate the velocity again
getting
But notice this is the same as 
(see eq. 5.7).
That says that the acceleration points in the opposite direction
to the radius vector 
. The acceleration is pointing towards
the center of the circle. Its magnitude is just 
.
But from eq. 5.9 this just says, da da:
This is called the centripetal acceleration.
