Lecture 9: Parametric Equations

Objectives

• Be able to use parametric equations to describe the motion of a point

• Be able to find the arclength of a curve described by parametric equations as well as the area underneath such a curve.

• Know how to find the curvature of curves in 2D and 3D.

Corresponding Sections in Simmons 17.1,17.2,17.5

Parametric Equations

• It is often easier to express both x and y as a function of another variable (such as the time t or an angle ) instead of writing y in terms of x.

• This new variable is called a parameter and these are called parametric equations.

• Can be thought of as a map:

Example: Projectile motion

• We can express y in terms of x by eliminating t from these equations, giving

• However, the parametric equations are cleaner and much more informative.

Example: Circular Motion

• Consider a particle moving counterclockwise around a circle or radius r centered around the origin at constant speed.

Velocity and Acceleration• Given parametric equations , • Example: For projectile motion with , and • Example: For circular motion with, and

Arclength with Parametric Equations

• If we have function x(t), y(t), the length of the segment from t to t+dt is

• More generally,

Example: Arclength of a Circle

• Example: Arclength of a circle• , • ,

Area with Parametric Equations

• Area under a curve can also be done with parametric equations

• Example: What is the area under the curve described by , from to ?

Curvature

• The curvature of a curve is 1 divided by the radius of the circle which best approximates it at that point.

Recall the equations for circular motion: , and

• For a circle of radius R, (directed towards the center of the circle)

Curvature Cont.

• For a circle, so • In general, we want to take the component of

the acceleration which is perpendicular to the velocity and divide it by

Curvature of a curve

• In the special case where we have a curve , taking , , ,

• Example: The curve has curvature .