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

Jul 17, 2017

The speed is $= \frac{\sqrt{6} - \sqrt{2}}{2} = 0.52$

#### Explanation:

The speed is the derivative of the position

$p \left(t\right) = \sin \left(2 t - \frac{\pi}{4}\right) + 2$

$v \left(t\right) = p ' \left(t\right) = 2 \cos \left(2 t - \frac{\pi}{4}\right)$

When $t = \frac{\pi}{3}$

$v \left(\frac{\pi}{3}\right) = 2 \cos \left(2 \cdot \frac{\pi}{3} - \frac{\pi}{4}\right)$

$= 2 \cos \left(\frac{2}{3} \pi - \frac{1}{4} \pi\right)$

$= 2 \cdot \left(\cos \left(\frac{2}{3} \pi\right) \cdot \cos \left(\frac{\pi}{4}\right) + \sin \left(\frac{2}{3} \pi\right) \cdot \sin \left(\frac{1}{4} \pi\right)\right)$

$= 2 \cdot \left(- \frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}\right)$

$= \frac{\sqrt{6} - \sqrt{2}}{2} = 0.52$