Step $2$

We have determined that $v(t)=t^{2}6tC.$ We must now determine the value of $C.$ We are given that $v(0)=-16.$ Using this allows us to solve for $C$ as follows.

$v(t)=t^{2}6tC$

$v(0)=0^{2}6(0)C$

$-16=C$

Substituting $C=-16$ into the velocity function gives the following result $($in $\frac{m}{s}).$

$v(t)=$

$\frac{m}{s}$

Step $3$

$($b$)$ Find the distance traveled during the given time interval.

Recall that the total distance traveled from time $t_1$ to time $t_2$ is given by the following.

${\displaystyle \underset{h_1}{\overset{t_2}{}}}|v(t)|dt=$ total distance traveled

In a previous step we determined that $(t)=t^{2}6t-16.$ We wish to find the total distance traveled over the interval $0t5.$ Therefore, we must calculate the following.

${\displaystyle \underset{0}{\overset{5}{}}}{x}^{1}dt$

$$

SKIP $($YOU CANNOT COME BACK$)$