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

Dec 31, 2017

The average speed is $= 1.5 m {s}^{-} 1$

#### Explanation:

The speed is the integral of the acceleration

$a \left(t\right) = t$

Therefore,

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

Plugging in the initial conditions

$v \left(0\right) = 0$

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

Therefore,

$v \left(t\right) = {t}^{2} / 2$

The average speed in the interval $t \in \left[0 , 3\right]$ is

$3 \overline{v} = {\int}_{0}^{3} {t}^{2} / 2 \mathrm{dt} = {\left[{t}^{3} / 6\right]}_{0}^{3} = \left(\frac{27}{6}\right) - \left(0\right)$

$\overline{v} = \frac{9}{6} = \frac{3}{2} = 1.5 m {s}^{-} 1$