Parametric and Polar Curves: Velocity, 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.

Figure %: Lindsay's Position, (x(t), y(t)), in the Coordinate Plane

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 t₀ are simply

(x'(t₀), y'(t₀))

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.

Figure %: The Velocity Vector for Lindsay

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 t₀,

(x''(t₀), y''(t₀))

is called the acceleration vector of the original curve at time t₀. 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.