# What is the average speed of an object that is moving at 5 m/s at t=0 and accelerates at a rate of a(t) =20-t^2 on t in [1,5]?

Apr 4, 2017

The average speed is $= 52 m {s}^{-} 1$

#### Explanation:

The velocity is the integral of the acceleration.

$a \left(t\right) = 20 - {t}^{2}$

$v \left(t\right) = \int \left(20 - {t}^{2}\right) \mathrm{dt}$

$= 20 t - {t}^{3} / 3 + C$

We apply the initial conditions to find $C$

$v \left(0\right) = 5$

Therefore, $C = 5$

So,

$v \left(t\right) = 20 t - {t}^{3} / 3 + 5$

We can calculate the average value of the velocity

$\left(5 - 1\right) \overline{v} = {\int}_{1}^{5} \left(20 t - {t}^{3} / 3 + 5\right) \mathrm{dt}$

$= {\left[\frac{20}{2} {t}^{2} - {t}^{4} / 12 + 5 t\right]}_{1}^{5}$

$= \left(250 - \frac{625}{12} + 25\right) - \left(10 - \frac{1}{12} + 5\right)$

$= \left(\frac{2675}{12}\right) - \left(\frac{179}{12}\right)$

$= 208$

So,

$\overline{v} = \frac{208}{4} = 52 m {s}^{-} 1$