# What is the average speed of an object that is moving at 22 m/s at t=0 and accelerates at a rate of a(t) =t-8 on t in [0,2]?

Mar 17, 2018

The average speed is $= 3.67 m {s}^{-} 1$

#### Explanation:

$\text{Average speed " = "total distance travelled" /" time taken to cover that distance}$

The acceleration is

$a \left(t\right) = t - 8$

The velocity is

$v \left(t\right) = \int a \left(t\right) \mathrm{dt} = \int \left(t - 8\right) \mathrm{dt} = \frac{1}{2} {t}^{2} - 8 t + C$

Plugging in the initial conditions

$v \left(0\right) = 22 m {s}^{-} 1$

$22 = 0 - 0 + C$

Therefore,

$v \left(t\right) = \frac{1}{2} {t}^{2} - 8 t + 22$

The average speed is

$\overline{v} = \frac{1}{2 - 0} {\int}_{0}^{2} \left(\frac{1}{2} {t}^{2} - 8 t + 22\right) \mathrm{dt}$

$= \frac{1}{2} {\left[\frac{1}{6} {t}^{3} - 4 {t}^{2} + 22 t\right]}_{0}^{2}$

$= \frac{1}{2} \left(\left(\frac{1}{6} \cdot 8 - 16 + 22\right) - \left(0\right)\right)$

$= 3.67 m {s}^{-} 1$