# The position of an object moving along a line is given by p(t) = cos(t- pi /3) +1 . What is the speed of the object at t = (2pi) /4 ?

Jul 12, 2016

$v \left(\frac{2 \pi}{4}\right) = - \frac{1}{2}$

#### Explanation:

Since the equation given for the position is known, we can determine an equation for the velocity of the object by differentiating the given equation:

$v \left(t\right) = \frac{d}{\mathrm{dt}} p \left(t\right) = - \sin \left(t - \frac{\pi}{3}\right)$

plugging in the point at which we want to know speed:
$v \left(\frac{2 \pi}{4}\right) = - \sin \left(\frac{2 \pi}{4} - \frac{\pi}{3}\right) = - \sin \left(\frac{\pi}{6}\right) = - \frac{1}{2}$

Technically, it might be stated that the speed of the object is, in fact, $\frac{1}{2}$, since speed is a directionless magnitude, but I have chosen to leave the sign.