Answer
$y=-0.1265sin(\sqrt{10}t)-0.09cos(\sqrt{10}t)$
$0.1552m$
Work Step by Step
We can determine the required equation and the maximum upward displacement as follows:
$y=\frac{v_{\circ}}{\omega_n}sin(\omega_n t)+y_{\circ} cos(\omega_n)t$eq(1)
We know that
$\omega_n=\sqrt{\frac{K}{m}}$
$\implies \omega_n=\sqrt{\frac{80}{8}}=\sqrt{10}rad/s$
We plug in the known values in eq(1) to obtain:
$y=-\frac{0.4}{\sqrt{10}}sin(\sqrt{10}t)-0.09cos(\sqrt{10}t)$
$\implies y=-0.1265sin(\sqrt{10}t)-0.09cos(\sqrt{10}t)$
Now we find the maximum upward displacement
$C=\sqrt{(\frac{v_{\circ}}{\omega_n})^2+(y_{\circ})^2}$
We plug in the known values to obtain:
$C=\sqrt{(\frac{0.4}{\sqrt{10}})^2+(0.09)^2}$
This simplifies to:
$C=0.1552m$
