# What is the average speed of an object that is still at t=0 and accelerates at a rate of a(t) =3t-4 from t in [2, 3]?

Jul 16, 2017

The average speed is $= - 0.5 m {s}^{-} 1$

#### Explanation:

The speed is the integral of the acceleration

$a \left(t\right) = 3 t - 4$

$v \left(t\right) = \int \left(3 t - 4\right) \mathrm{dt} = \frac{3}{2} {t}^{2} - 4 t + C$

Plugging in the initial conditions at $t = 0$

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

The average velocity is

$\left(3 - 2\right) \overline{v} = {\int}_{2}^{3} \left(\frac{3}{2} {t}^{2} - 4 t\right) \mathrm{dt}$

$= {\left[\frac{1}{2} {t}^{3} - 2 {t}^{2}\right]}_{2}^{3}$

$= \left(\frac{27}{2} - 18\right) - \left(4 - 8\right)$

$\overline{v} = - 0.5$