# What is the average speed of an object that is moving at 4 m/s at t=0 and accelerates at a rate of a(t) =2t^2-2t+4 on t in [0,3]?

Mar 16, 2016

${v}_{a} = 11 , 5 \text{ } \frac{m}{s}$

#### Explanation:

$v \left(t\right) = \int \left(2 {t}^{2} - 2 t + 4\right) d t$
$v \left(t\right) = \frac{2}{3} {t}^{3} - \frac{2}{2} {t}^{2} + 4 t + C \text{ for t=0 v=4 m/s C=4}$
$v \left(t\right) = \frac{2}{3} {t}^{3} - {t}^{2} + 4 t + 4$
$\Delta x = {\int}_{0}^{3} \left(\frac{2}{3} {t}^{3} - {t}^{2} + 4 t + 4\right) d t$
$\Delta x = | \frac{2}{3} \cdot \frac{1}{4} {t}^{4} - \frac{1}{3} {t}^{3} + \frac{4}{2} {t}^{2} + 4 t {|}_{0}^{3}$
$\Delta x = \left(\frac{1}{6} \cdot 81 - \frac{27}{3} + 2 \cdot 9 + 4 \cdot 3\right) - 0$
$\Delta x = \frac{27}{2} - 9 + 18 + 12 \text{ } \Delta x = \frac{27}{2} + 21$
$\Delta x = \frac{42 + 27}{2}$
$\Delta x = \frac{69}{2} \text{ } \Delta x = 34 , 5 m$
$\Delta x \text{:displacement at interval (0,3)}$
${v}_{a} = \frac{\Delta x}{\Delta t}$
${v}_{a} = \frac{34 , 5}{3 - 0}$
${v}_{a} = \frac{34 , 5}{3}$
${v}_{a} = 11 , 5 \text{ } \frac{m}{s}$