Answer
$y=(17.27sin(13.9t)-2cos(13.9t))in$
$C=17.4in$
Work Step by Step
To determine the equation of motion:
$y=\frac{v_{o}}{ω_{n}}sin(ω_{n}t)+y_{o}cos(ω_{n}t)...eq(1)$
The natural frequency can be calculated by:
$ω_{n}=\sqrt \frac{K}{m}$
$ω_{n}=\sqrt \frac{3lb/in\times12in/ft}{6/32.2}=13.9 rad/s$
By substituting $ω_{n}$ in $eq(1)$ we obtain the equation describes the motion $eq(2)$:
$y=\frac{20ft/s\times12in/ft}{13.9rad/s}sin(13.9t)+(-2in)cos(13.9t)$
$y=(17.27sin(13.9t)-2cos(13.9t))in...eq(2)$
To determine the maximum displacement $C$:
$C=\sqrt {\frac {v_{o}}{ω_{n}}^{2}+y_{o}^{2}}$
$C=\sqrt {(\frac {20ft/s\times12in/ft}{13.9rad/s})^{2}+(-2in)^{2}}$
$C=17.4in$
