# The position of an object moving along a line is given by p(t) = 3t - tsin(( pi )/8t) . What is the speed of the object at t = 2 ?

Feb 22, 2018

The speed is $= 1.74 m {s}^{-} 1$

#### Explanation:

Reminder :

The derivative of a product

$\left(u v\right) ' = u ' v - u v '$

$\left(t \sin \left(\frac{\pi}{8} t\right)\right) ' = 1 \cdot \sin \left(\frac{\pi}{8} t\right) + \frac{\pi}{8} t \cos \left(\frac{\pi}{8} t\right)$

The position of the object is

$p \left(t\right) = 3 t - t \sin \left(\frac{\pi}{8} t\right)$

The speed of the object is the derivative of the position

$v \left(t\right) = p ' \left(t\right) = 3 - \sin \left(\frac{\pi}{8} t\right) - \frac{\pi}{8} t \cos \left(\frac{\pi}{8} t\right)$

When $t = 2$

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

$= 3 - \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{8} \pi$

$= 1.74 m {s}^{-} 1$