NOTE: Provide thorough derivations or explanations, and

computer generated, fully annotated graphs for full credit

$($continuing from Problem $3$ of HW #$1)$ Let us call $v_A$ and $v_B$ the velocities and $a_A$ and $a_B$ the

accelerations of the floats A and $B,$ respectively $($velocity or acceleration are positive pointing

upwards$)$ :

$($a$)$ Find $v_A,v_B,a_A$ and $a_B$ at the time $t=0.(8$ points$)$

$($b$)$ Find $v_A,v_B,a_A$ and $a_B$ at the time the wave elevations of the floats A and B are equal in

value but opposite in sign $($the same time as what you determined in part $($e$)$ of Problem $3$ of

HW#$1).(12$ points$)$

A sinusoidal wave is passing under a lifeboat which is stranded in the ocean. If the maximum

$($vertical$)$ velocity of the life$-$boat is $1\mathrm{.}5\frac{m}{sec}$ and the maximum $($vertical$)$ acceleration is $2\mathrm{.}3$

$\frac{m}{se{c}^2},$ find the height, the period, and the speed of this wave $(12$ points$).$ Then determine the

maximum slope angle of the boat $($in degrees$),$ assuming that its slope is matching the slope of

the wave profile $(6$ points$).$ We assume that the life$-$boat moves up and down $($without moving in

the horizontal direction$)$ like a float. Additionally, we assume that the relationship $L=\frac{g{T}^2}{2},$ as

mentioned in HW#$1,$ for deep water waves, applies.

Using Taylor expansion, prove that for $k>0$ and for the following approximations

apply: $(12$ points$)$: $($ Reminder: $\frac{d{e}^x}{dx}=e^x).$

$e^{kz}$~~$1+kz$

$(1+z{)}^{\frac{1}{3}}$~~$1+\frac{z}{3}$