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

May 10, 2016

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

#### Explanation:

$v \left(t\right) = \frac{d}{d t} p \left(t\right)$

$v \left(t\right) = \frac{d}{d t} \left(\sin \left(2 t - \frac{\pi}{3}\right) + 2\right)$

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

$\text{for } t = \left(\frac{2 \pi}{3}\right) \rightarrow v \left(\frac{2 \pi}{3}\right) = 2 \cdot \cos \left(2 \cdot \frac{2 \pi}{3} - \frac{\pi}{3}\right)$

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

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

$\cos \pi = - 1$

$v \left(\frac{2 \pi}{3}\right) = - 2 \cdot 1$

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