# What is the average speed of an object that is moving at 9 m/s at t=0 and accelerates at a rate of a(t) =2t-5 on t in [1,4]?

Apr 1, 2017

The average speed is $= 3.5 m {s}^{-} 2$

#### Explanation:

$a \left(t\right) = 2 t - 5$

The speed is the integral of the acceleration

$v \left(t\right) = \int \left(2 t - 5\right) \mathrm{dt}$

$v \left(t\right) = \frac{2}{2} {t}^{2} - 5 t + C$

Initial conditions are,

$v \left(0\right) = 9$

Therefore,

$v \left(0\right) = 0 - 0 + C = 9$

So,

$v \left(t\right) = {t}^{2} - 5 t + 9$

The average velocity is $\overline{v}$

$\left(4 - 1\right) \overline{v} = {\int}_{1}^{4} \left({t}^{2} - 5 t + 9\right) \mathrm{dt}$

$3 \overline{v} = {\left[{t}^{3} / 3 - \frac{5}{2} {t}^{2} + 9 t\right]}_{1}^{4}$

$= \left(\frac{64}{3} - 40 + 36\right) - \left(\frac{1}{3} - \frac{5}{2} + 9\right)$

$3 \overline{v} = \frac{52}{3} - \frac{41}{6} = \frac{63}{6} = \frac{21}{2}$

$\overline{v} = \frac{7}{2} m {s}^{-} 1$