# The position of an object moving along a line is given by p(t) = sin(3t- pi /4) +2 . What is the speed of the object at t = (3pi) /4 ?

Jun 4, 2016

Velocity of an object is the time derivative of it's position coordinate(s). If the position is given as a function of time, first we must find the time derivative to find the velocity function.

#### Explanation:

We have $p \left(t\right) = S \in \left(3 t - \frac{\pi}{4}\right) + 2$

Differentiating the expression,

$\frac{\mathrm{dp}}{\mathrm{dt}} = \frac{d}{\mathrm{dt}} \left[S \in \left(3 t - \frac{\pi}{4}\right) + 2\right]$

$p \left(t\right)$ denotes position and not momentum of the object. I clarified this because $\vec{p}$ symbolically denotes the momentum in most cases.

Now, by definition, $\frac{\mathrm{dp}}{\mathrm{dt}} = v \left(t\right)$ which is the velocity. [or in this case the speed because the vector components are not given].

Thus, $v \left(t\right) = C o s \left(3 t - \frac{\pi}{4}\right) . \frac{d}{\mathrm{dt}} \left(3 t - \frac{\pi}{4}\right)$
$\implies v \left(t\right) = 3 C o s \left(3 t - \frac{\pi}{4}\right)$

At $t = \frac{3 \pi}{4}$

$v \left(\frac{3 \pi}{4}\right) = 3 C o s \left(3. \frac{3 \pi}{4} - \frac{\pi}{4}\right)$

$\implies$ Speed $= 3 C o s 2 \pi = 3$ units.