What is the average speed of an object that is moving at 12 m/s at t=0 and accelerates at a rate of a(t) =7-t on t in [0,4]?

Mar 3, 2016

I found $5 \frac{m}{s}$

Explanation:

I do not want to complicate it too much but I would integrate the acceleration to go back to the expression of velocity:
$v \left(t\right) = \int a \left(t\right) \mathrm{dt} = \int \left(7 - t\right) \mathrm{dt} = 7 t - {t}^{2} / 2 + c$
now we get the constant $c$ by imposing that at $t = 0$ we have $v \left(0\right) = 12 \frac{m}{s}$. So:
$12 = \left(7 \cdot 0\right) - {\left(0\right)}^{2} / 2 + c$
so that $c = 12 \frac{m}{s}$
so the velocity will be:
$v \left(t\right) = 7 t - {t}^{2} / 2 + 12$
that at $t = 4 s$ is:
$v \left(4\right) = \left(7 \cdot 4\right) - \frac{{4}^{2}}{2} + 12 = 32 \frac{m}{s}$

Average speed will be: $\frac{\Delta v}{\Delta t} = \frac{v \left(4\right) - v \left(0\right)}{4 - 0} = \frac{32 - 12}{4} = 5 \frac{m}{s}$