# 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.

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. 