# What is the average speed of an object that is moving at 3 m/s at t=0 and accelerates at a rate of a(t) =t^2-1 on t in [0,2]?

May 8, 2017

The average speed is $= 2.67 m {s}^{-} 1$

#### Explanation:

The speed is the integral of the acceleration

$a \left(t\right) = {t}^{2} - 1$

$v \left(t\right) = \int \left({t}^{2} - 1\right) \mathrm{dt}$

$v \left(t\right) = \frac{1}{3} {t}^{3} - t + C$

Plugging in the initial conditions

$v \left(0\right) = 0 - 0 + C = 3$

So,

$C = 3$

and

$v \left(t\right) = \frac{1}{3} {t}^{3} - t + 3$

The average speed is

$2 \overline{v} = {\int}_{0}^{2} \left(\frac{1}{3} {t}^{3} - t + 3\right) \mathrm{dt}$

$= {\left[\frac{1}{12} {t}^{4} - \frac{1}{2} {t}^{2} + 3 t\right]}_{0}^{2}$

$= \left(\frac{16}{12} - \frac{4}{2} + 6\right) - \left(0\right)$

$= \frac{16}{3}$

$\overline{v} = \frac{\frac{16}{3}}{2} = 2.67 m {s}^{-} 1$