# What is the average speed of an object that is moving at 15 m/s at t=0 and accelerates at a rate of a(t) =7-2t^2 on t in [0,3]?

Mar 6, 2016

${v}_{a} = \frac{81}{3} = 27 \text{ } \frac{m}{s}$

#### Explanation:

v(t)=int(a(t)d t
$v \left(t\right) = \int \left(7 - 2 {t}^{2}\right) d t \text{ } v \left(t\right) = 7 x - \frac{2}{3} {t}^{3} + C$
$\mathmr{if} t = 0 \text{ "v=15 m/s" } v \left(t\right) = 7 x - \frac{2}{3} {t}^{3} + 15$
$\Delta x = {\int}_{0}^{3} v \left(t\right) d t \text{ } \Delta x = {\int}_{0}^{3} \left(7 x - \frac{2}{3} {t}^{3} + 15\right) d t$
$\Delta x = | \frac{7}{2} {x}^{2} - \frac{2}{12} {t}^{4} + 15 t {|}_{0}^{3}$
$\Delta x = \frac{7}{2} \cdot {3}^{2} - \frac{2}{12} \cdot {3}^{4} + 15 \cdot 3$
$\Delta x = \frac{63}{2} - \frac{81}{6} + 45 \text{ } \Delta x = \frac{63}{2} - \frac{27}{2} + 45$
$\Delta x = 36 + 45 \text{ } \Delta x = 81$
${v}_{a} = \frac{81}{3} = 27 \text{ } \frac{m}{s}$