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

May 3, 2017

The speed is $= - 0.33 m {s}^{-} 1$

#### Explanation:

The speed is the derivative of the position.

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

$v \left(t\right) = p ' \left(t\right) = 1 - \sin \left(\frac{\pi}{4} t\right) - \frac{\pi}{4} t \cos \left(\frac{\pi}{4} t\right)$

When $t = 1$

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

$= 1 - \frac{\sqrt{2}}{2} - \frac{\pi}{4} \cdot \frac{\sqrt{2}}{2}$

$= 1 - 0.707 - 0.555$

$= - 0.33$