SummaryVelocity, Acceleration, and Parametric Curves

Suppose an ice skater named Lindsay is gliding around on a frozen coordinate plane.
Define functions x(t), y(t), so that at time t (in seconds) Lindsay's position on the
coordinate plane is given by (x(t), y(t)). If Lindsay starts at time t = 0 and stops at
time t = 15, she will trace out the parametric curve consisting of the points (x(t), y(t))
with t in the interval [0, 15], perhaps like the one sketched below.

Two questions naturally arise. First, what is Lindsay's velocity (direction and speed) at
any given moment? Second, how is it changing (that is, what is her acceleration)? It is
actually fairly easy to answer these questions using the derivative.

Lindsay's velocity at time t can be represented by an arrow with a certain direction and
length, called the velocity vector. The direction of the vector will indicate her
instantaneous direction, and the length of the vector will equal her instantaneous speed.
We would expect the direction of the velocity vector at time t to be the same as the
direction of the tangent line at (x(t), y(t)) to the curve describing Lindsay's path. If
we imagine the velocity vector to have its tail at the origin of a coordinate plane (rather
than at a point of tangency to the parametric curve), we need only specify the coordinates
of its head to give its length and direction.

It is fairly easy to see that the correct coordinates for the head of the velocity vector at
time t_{0} are simply

(x'(t_{0}), y'(t_{0}))

where the derivatives are with respect to t. The speed is equal to the length of this
velocity vector:

This is all illustrated in the figure below.

Note that the head of the velocity vector (with tail fixed at the origin), located at
(x'(t), y'(t)) at time t, traces out another parametric curve. The velocity vector of
this new curve at time t_{0},

(x''(t_{0}), y''(t_{0}))

is called the acceleration vector of the original curve at time t_{0}. Its direction
indicates the instantaneous direction of motion of the head of the velocity vector and its
length is the speed at which the head of the velocity vector is moving.

It is important to realize that there is a difference between the length of the
acceleration vector, which is the rate at which the velocity it changing, and the rate at
which the speed is changing. For instance, if an object is moving in the circular path
(x(t), y(t)) = (cos(t), sin(t)), then the speed is constant (for (x'(t), y'(t)) = (- sin(t), cos(t)) has length = 1, yet the acceleration is
nonzero and has constant length -- (x''(t), y''(t)) = (- cos(t), - sin(t)) has length
= 1. One can think of this phenomenon as follows: the
acceleration contains contributions from the rate of change of the speed, and from the rate
of change of the direction of the velocity.