# The position of an object moving along a line is given by p(t) = 4t - tsin(( pi )/8t) . What is the speed of the object at t = 3 ?

Jun 20, 2016

In one dimension, speed is just the magnitude of velocity, such that if we had a negative value we would just take the positive version.

To find the speed function, we will need to differentiate the position function with respect to t:

Let $s \left(t\right)$ be the speed function:
$s \left(t\right) = 4 - \sin \left(\frac{\pi}{8} t\right) - \frac{\pi}{8} t \cos \left(\frac{\pi}{8} t\right)$
(I've assumed proficiency with the product and chain rule)

Therefore the speed at $t = 3$ is given by:
$s \left(3\right) = 4 - \sin \left(3 \frac{\pi}{8}\right) - 3 \frac{\pi}{8} \cos \left(3 \frac{\pi}{8}\right)$
$s \left(3\right) = 2.63 m {s}^{-} 1$ (ensuring the take the trig functions in radians)