Answer
The transverse speed is $~~4.24~m/s$
Work Step by Step
$y(x,t) = (15.0~cm)~cos(\pi x-\omega t)$
We can consider $x=0$ and find a value $t$ when $y(x,t) = 12.0~cm$:
$y(x,t) = (15.0~cm)~cos(\pi x-15 \pi t)$
$y(0,t) = (15.0~cm)~cos(0-15 \pi t) = 12.0~cm$
$cos(-15 \pi t) = 0.80$
$-15 \pi t = -0.6435$
$t = \frac{0.6435}{15 \pi}$
We can find the transverse velocity at this time:
$u = (15.0~cm)(15 \pi)~sin(\pi x-15 \pi t)$
$u = (15.0~cm)(15 \pi)~sin(0-15 \pi \cdot 0.6435/15\pi)$
$u = (15.0~cm)(15 \pi)~sin(-0.6435)$
$u = -424~cm/s$
$u = -4.24~m/s$
The transverse speed is $~~4.24~m/s$
