Single Variable Calculus, Chapter 3, 3.7, Section 3.7, Problem 8

Suppose that a ball is given a push so that it has an initial velocity of $\displaystyle 5 \frac{m}{s}$ down a certain inclined plane, then the distance it has rolled after $t$ seconds is $s = 5t + 3t^2$

a.) Find the velocity after $2s$
b.) How long does it take for the velocity to reach $\displaystyle 35 \frac{m}{s}$



$
\begin{equation}
\begin{aligned}
\text{a.) velocity } &= s'(t) = \frac{ds}{dt}\\
\\
&= 5 \frac{d}{dt} (t) + 3 \frac{d}{dt} (t^2)\\
\\
&= 5 (1) + 3(2t)\\
\\
&= 5 + 6t \frac{m}{s}\\
\end{aligned}
\end{equation}
$

The velocity after $2s$ is $\displaystyle v(2) = 5 + 6(2) = 17 \frac{m}{s}$

b.) if $\displaystyle v = 35 \frac{m}{s}$, solving for $t$


$
\begin{equation}
\begin{aligned}
35 &= 5 + 6t\\
t &= 5 \text{ seconds}
\end{aligned}
\end{equation}
$