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Velocity, Acceleration, and Parametric Curves

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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 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.