# 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) =3-t on t in [0,5]?

May 7, 2018

The average speed is $= 12.33 m {s}^{-} 1$

#### Explanation:

The acceleration is

$a \left(t\right) = 3 - t$

The velocity is the integral of the acceleration

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

Plugging in the initial conditions

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

Therefore,

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

$C = 9$

$v \left(t\right) = 3 t - \frac{1}{2} {t}^{2} + 9$

The average speed on $t \in \left[0 , 5\right]$ is

$\left(5 - 0\right) \overline{v} = {\int}_{0}^{5} \left(3 t - \frac{1}{2} {t}^{2} + 9\right) \mathrm{dt}$

$\left(5\right) \overline{v} = {\left[\frac{3}{2} {t}^{2} - \frac{1}{6} {t}^{3} + 9 t\right]}_{0}^{5}$

$= \left(\frac{75}{2} - \frac{125}{6} + 45\right) - \left(0\right)$

$= \frac{185}{3}$

$\overline{v} = \frac{185}{15} = 12.33 m {s}^{-} 1$