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

May 20, 2016

$s = 3 + \frac{\pi \sqrt{3}}{6} \cong 3.91$

#### Explanation:

The speed of an object is the rate of change of it's position with respect to time - in other words, the magnitude of the derivative of the position with respect to time:

$s = | \frac{d}{\mathrm{dt}} p \left(t\right) | = | \dot{p} \left(t\right) |$

In our case we need to do the derivative of our function:

$\dot{p} \left(t\right) = \frac{d}{\mathrm{dt}} p \left(t\right) = 3 + \frac{\pi}{3} \sin \left(\frac{\pi}{3} t\right)$

plugging in the time given

$\dot{p} \left(2\right) = 3 + \frac{\pi}{3} \sin \left(2 \frac{\pi}{3}\right) = 3 + \frac{\pi \sqrt{3}}{6}$

Therefore the speed, being the absolute value of this, is the same:

$s = 3 + \frac{\pi \sqrt{3}}{6} \cong 3.91$