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 .
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