# Let a particle be moving according to the following parametric equation. Find all times(including...

## Question:

Let a particle be moving according to the following parametric equation. Find all times(including negative times) when the particle is moving in the clockwise directions. Find speed. Assume {eq}t{/eq} is in seconds. {eq}x(t)= 5\cos(\sin t); y(t)= 5\sin(\sin t){/eq}

## Motion:

We often use space curves, i.e. vector parametric curves {eq}\vec r (t) {/eq}, to describe the trajectories of objects. When we do so the derivative of the space curve is the object's velocity function, and the magnitude of the velocity is the speed.

## Answer and Explanation:

The position function is

{eq}\begin{align*} \vec r (t) &= \left< 5\cos(\sin t), 5\sin(\sin t) \right> \end{align*} {/eq}

so the velocity is

{eq}\begin{align*} \vec r' (t) &= \left< -5 \sin (\sin t)\ \cos t,\ 5\cos(\sin t)\ \cos t \right> \end{align*} {/eq}

And the speed is

{eq}\begin{align*} | \vec r' (t) | &= \sqrt{(-5 \sin (\sin t)\ \cos t)^2 +(5\cos(\sin t)\ \cos t )^2} \\ &= 5 | \cos t | \sqrt{\sin^2 (\sin t) + \cos^2 (\sin t)} \\ &= 5 | \cos t | \end{align*} {/eq}

Note that when {eq}t = 0 {/eq} we are at the point {eq}(5,0) {/eq}. From here we move up the arc to the top at {eq}t = \frac\pi2 {/eq}: {eq}(5 \cos 1, 5 \sin 1) {/eq}. The turn happens here, which means this is when we start moving clockwise. We go all the way down the arc to the point where {eq}t = \frac{3\pi}2 {/eq}: {eq}(5 \cos 1, -5 \sin 1) {/eq}. From here back up (so counterclockwise again) then back down. So we are clockwise on the interval

{eq}\begin{align*} \left( \frac\pi2, \frac{3\pi}2 \right) \cup \left( \frac{5\pi}2, \frac{7\pi}2 \right) \cdots \end{align*} {/eq}

where we use the ellipses to indicate that this pattern goes on forever, we can also extend it backwards:

{eq}\begin{align*} \cdots \left( - \frac{7\pi}2, -\frac{5\pi}2 \right) \cup \left( -\frac{3\pi}2, -\frac{\pi}2 \right) \cup \left( \frac\pi2, \frac{3\pi}2 \right) \cup \left( \frac{5\pi}2, \frac{7\pi}2 \right) \cdots \end{align*} {/eq}

So the particle moves clockwise on an infinite number of time intervals, all following the pattern above. This set is the set of times for which movement is counterclockwise.

#### Learn more about this topic:

from Precalculus: High School

Chapter 24 / Lesson 3