# The position of an object moving along a line is given by p(t) = 2t^3 - 2t^2 +1. What is the speed of the object at t = 3 ?

Aug 11, 2017

$v = 42 \text{ units}$

#### Explanation:

The speed of an object is the magnitude of the object's velocity , which is the derivative of displacement (magnitude is position).

To find the speed of the object at time $t$, we can begin by taking the derivation of the provided equation for position:

$p \left(t\right) = 2 {t}^{3} - 2 {t}^{2} + 1$

$\implies p ' \left(t\right) = v \left(t\right) = 6 {t}^{2} - 4 t$

At $t = 3$, we have:

$v \left(t\right) = 6 {\left(3\right)}^{2} - 4 \left(3\right)$

$= 54 - 12$

$= 42$

$\therefore$ At $t = 3$, the object has a speed of $42 \text{ units.}$