Study guide and
3 practice problems
on:
Speed of a parameterization
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If $\bfx(t)$ is the parameterization of a curve, the speed at $t$ is given by the magnitude of the velocity: $$| \bfv | = \left |\frac{d \bfx}{d t} \right|.$$
Related topics
Parameterized curves
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Lines, Planes, and Curves
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Multivariable calculus
(147 problems)
Practice problems
A frisbee of radius $r$ translates rightward at speed $v$ meter/second. It rotates clockwise at $\omega$ radian/second. Initially the frisbee is centered at the origin.
Find the trajectory swept by the point initially at $(0,r)$.
Compute the speed as a function of time.
Describe when the speed is largest and smallest.
Solution
Find the arc length of the helix $x(t) = \cos t, y(t) = \sin t, z(t) = t$ traced from $t=1$ to $t=2$.
Solution
An ant is on a merry-go-ground that is rotating clockwise at $\omega$ radians per second. Initially, the ant is at $(R,0)$. From the ant's perspective, it walks toward the center with speed $v$. Several snapshots in time are as follows:
Find the parameterization of the path taken by the ant (relative to the ground)
Compute the speed of the ant as a function of $t$. When is it largest?
Set up, but do not evaluate, an integral for the arc length of the path taken by the ant between $t=0$ and when the ant reaches the origin
Solution