# What is the speed of the particle?

## A particle moves with its position given by x=cos(4t) and y=sin(t), where positions are given in feet from the origin and time t is in seconds.

Nov 7, 2017

Oh. Oh. Oh. I got this one.

#### Explanation:

You can find the velocity by adding up the components, which you find by taking the first derivative of the x & y functions:

$\frac{\mathrm{dx}}{\mathrm{dt}} = - 4 \sin \left(4 t\right)$
$\frac{\mathrm{dy}}{\mathrm{dt}} = \cos \left(t\right)$

So, your velocity is a vector with components as given above.

The speed is the magnitude of this vector, which can be found via the Pythagorean theorem:

$s = \sqrt{{\left(- 4 \sin \left(4 t\right)\right)}^{2} + {\cos}^{2} \left(t\right)}$

...there may be some clever way to simplify this further, but perhaps this will do.