# What is the average speed, on t in [0,5], of an object that is moving at 10 m/s at t=0 and accelerates at a rate of a(t) =2t^2-10 on t in [0,3]?

Aug 6, 2017

The average speed is $= 1.6 m {s}^{-} 1$

#### Explanation:

The speed is the integral of the acceleration

$a \left(t\right) = 2 {t}^{2} - 10$

$v \left(t\right) = \frac{2}{3} {t}^{3} - 10 t + C$

Plugging in the initial conditions

$v \left(0\right) = 0 - 0 + C = 10$

Therefore,

$v \left(t\right) = \frac{2}{3} {t}^{3} - 10 t + 10$ on the interval $I = \left[0 , 3\right]$

The total distance travelled in $5 s$ is

$d = {\int}_{0}^{3} \left(\frac{2}{3} {t}^{3} - 10 t + 10\right) \mathrm{dt} + v \left(3\right) \cdot 2$

$= {\left[\frac{1}{3} {t}^{4} - 5 {t}^{2} + 10 t\right]}_{0}^{3} + \left(18 - 30 + 10\right) \cdot 2$

$= \left(27 - 45 + 30\right) + \left(- 4\right)$

$= 8$

The average speed is

$\overline{v} = \frac{8}{5} = 1.6 m {s}^{1}$