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+1 on t in [1,2]?

Mar 2, 2018

The average speed is $= 12.83 m {s}^{-} 1$

Explanation:

The speed is the integral of the acceleration

$a \left(t\right) = 2 t + 1$

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

$= {t}^{2} + t + C$

Plugging in the initial conditions

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

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

$\implies$, $C = 9$

Therefore,

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

The average speed is

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

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

$= \left(\frac{8}{3} + 2 + 18\right) - \left(\frac{1}{3} + \frac{1}{2} + 9\right)$

$= \frac{7}{3} + 11 - \frac{1}{2}$

$= 12.83 m {s}^{-} 1$