# What is the average speed of an object that is moving at 12 m/s at t=0 and accelerates at a rate of a(t) =2-3t^2 on t in [0,2]?

Jul 13, 2017

The average speed is $= 12 m {s}^{-} 1$

#### Explanation:

The speed is the integral of the acceleration

$a \left(t\right) = 2 - 3 {t}^{2}$

$v \left(t\right) = \int \left(2 - 3 {t}^{2}\right) \mathrm{dt} = 2 t - {t}^{3} + C$

Plugging in the initial conditions at $t = 0$

$v \left(0\right) = 0 - 0 + C = 12$, $\implies$, $C = 12$

So,

$v \left(t\right) = 2 t - {t}^{3} + 12$

The average speed is

$2 \overline{v} = {\int}_{0}^{2} \left(2 t - {t}^{3} + 12\right) \mathrm{dt}$

$= {\left[{t}^{2} - {t}^{4} / 4 + 12 t\right]}_{0}^{2}$

$= \left(4 - 4 + 24\right) - \left(0\right)$

$\overline{v} = \frac{24}{2} = 12$

The average speed is $\overline{v} = 12 m {s}^{-} 1$