Single Variable Calculus, Chapter 3, 3.4, Section 3.4, Problem 36

A elastic band with suspended mass that vibrates vertically has an equation motion of $s = 2 \cos t + 3 \sin t, t \geq 0$ where $s$ is measured in centimeters and $t$ in seconds. Assume that the positive direction is downward.

a.) Determine the velocity and acceleration at time $t$.
b.) Graph the velocity and acceleration functions.
c.) When does the mass pass through the equilibrium position for the first time?
d.) At what time is the speed the greatest?


$
\begin{equation}
\begin{aligned}
\text{a.) }& \text{velocity } = s' (t) &&= 2 \frac{d}{dt} \cos t + 3 \frac{d}{dt} \sin t\\
&&&= 2 (-\sin t)(1) + 3(\cos t)(1)\\
&&&= -2 \sin t + 3 \cos t\\
\\
\\
\phantom{x} &\text{acceleration } = s''(t) &&= -2\frac{d}{dt} \sin t + 3 \frac{d}{dt} \cos t\\
&&&= -2 (\cos t)(1)+3(-\sin t)(1)\\
&&&= -2\cos t - 3 \sin t
\end{aligned}
\end{equation}
$


b.)



c.) The mass is in equilibrium position whenever $s(t) = 0$ so,


$
\begin{equation}
\begin{aligned}
s & = 2 \cos t + 3 \sin t\\
0 & = 2 \cos t + 3 \sin t\\
t & = \tan^{-1} \left[ -\frac{2}{3}\right]\\
t & = -0.577 \text{ seconds, however } s(t) \text{ is defined only for } t\geq 0 \text{ so}\\
t & = -0.588 + \pi(n) \text{ ; where } \pi n \text{ corresponds to its succeeding period and } n \text{ is any integer.}
\end{aligned}
\end{equation}
$

But the problem asks for the first time that the mass pass through the equilibrium. So we have $n=1$,

$
\begin{equation}
\begin{aligned}
t & = - 0.588 + \pi(1)\\
t & = 2.5536 \text{ seconds}
\end{aligned}
\end{equation}
$

d.) based from the graph, the speed $s'(t)$ is greater at $t=2.5536 + \pi(n)$ seconds