# What is the average speed, on t in [0,5], of an object that is moving at 5 m/s at t=0 and accelerates at a rate of a(t) =5-2t on t in [0,4]?

Feb 16, 2018

The average speed is $= 9.53 m {s}^{-} 1$

#### Explanation:

The speed is the integral of the acceleration.

The acceleration is

$a \left(t\right) = 5 - 2 t$

The speed is

$v \left(t\right) = \int a \left(t\right) \mathrm{dt} = \int \left(5 - 2 t\right) \mathrm{dt}$

$= 5 t - {t}^{2} + C$

Plugging in the initial conditions

$v \left(0\right) = 5 \cdot 0 - 0 + C = 5$

Therefore,

$v \left(t\right) = 5 t - {t}^{2} + 5$

The speed when $t = 4$ is

$v \left(4\right) = 20 - 16 + 5 = 9 m {s}^{-} 1$

The average speed on the interval $\left[0 , 5\right]$ is

$\overline{v} \cdot \left(5 - 0\right) = {\int}_{0}^{4} \left(5 t - {t}^{2} + 5\right) \mathrm{dt} + 9 \cdot 1$

$5 \overline{v} = {\left[\frac{5}{2} {t}^{2} - \frac{1}{3} {t}^{3} + 5 t\right]}_{0}^{4} + 9$

$5 \overline{v} = \left(40 - \frac{64}{3} + 20\right) - \left(0\right) + 9 = \frac{116}{3} + 9 = 47.67$

$\overline{v} = \frac{47.67}{5} = 9.53 m {s}^{-} 1$