# The position of an object moving along a line is given by p(t) = 3t - 2sin(( pi )/8t) + 2 . What is the speed of the object at t = 24 ?

Sep 3, 2016

$v$ = $3.785$ $\frac{m}{s}$

#### Explanation:

The first time derivative of a position of an object gives the velocity of the object
$\dot{p} \left(t\right) = v \left(t\right)$
So, to get the velocity of the object we differentiate the position with respect to $t$
$p \left(t\right) = 3 t - 2 \sin \left(\frac{\pi}{8} t\right) + 2$
$\dot{p} \left(t\right) = 3 - 2 \cdot \frac{\pi}{8} \cdot \cos \left(\frac{\pi}{8} t\right) = v \left(t\right)$
So speed at $t = 24$ is
$v \left(t\right) = 3 - \frac{\pi}{4} \cos \left(\frac{\pi}{8} \cdot 24\right)$ ;or
$v \left(t\right) = 3 - \frac{\pi}{4} \left(- 1\right)$ ;or
$v \left(t\right) = 3 + \frac{\pi}{4} = 3.785$ $\frac{m}{s}$
Hence the speed of the object at $t = 24$ is $3.785$ $\frac{m}{s}$