b
The given wave equation is

\(y=3 \sin \frac{\pi}{2}(50 t-x)\)

\(y=3 \sin \left(25 \pi t-\frac{\pi}{2} x\right)\)                  \(...(i)\)

The standard wave equation is

\(y=A \sin (\omega t-k x)\)                                      \(...(ii)\)

Comparing \((i)\) and \((ii),\) we get 

\(\omega=25 \pi, \quad k=\frac{\pi}{2}\)

Wave velocity,

\(v=\frac{\omega}{k}=\frac{25 \pi}{(\pi / 2)}=50 \mathrm{ms}^{-1}\)

Particle velocity, \(v_{p}=\frac{d y}{d t}=\frac{d}{d t}\left(3 \sin \left(25 \pi t-\frac{\pi}{2}\right)\right)\)

\(=75 \pi \cos \left(25 \pi t-\frac{\pi}{2}\right)\)

Maximum particle velocity, \(\left(v_{p}\right)_{\max }=75 \pi \mathrm{ms}^{-1}\)

\(\therefore \frac{\left(v_{p}\right)_{\max }}{v}=\frac{75 \pi}{50}=\frac{3}{2} \pi\)