In the given progressive wave $y = 5 \sin(100\pi t + 0.4\pi x)$,where $y$ and $x$ are in $m$ and $t$ is in $s$. Find the:
$(a)$ Amplitude
$(b)$ Wavelength
$(c)$ Frequency
$(d)$ Wave velocity
$(e)$ Particle velocity amplitude.

(N/A) Comparing the given equation $y = 5 \sin(100\pi t + 0.4\pi x)$ with the standard wave equation $y = a \sin(\omega t + kx + \phi)$:
$(a)$ Amplitude $a = 5 \ m$.
$(b)$ Wave number $k = 0.4\pi$. Since $k = \frac{2\pi}{\lambda}$,we have $0.4\pi = \frac{2\pi}{\lambda} \Rightarrow \lambda = \frac{2}{0.4} = 5 \ m$.
$(c)$ Angular frequency $\omega = 100\pi$. Since $\omega = 2\pi f$,we have $100\pi = 2\pi f \Rightarrow f = 50 \ Hz$.
$(d)$ Wave velocity $v = \frac{\omega}{k} = \frac{100\pi}{0.4\pi} = 250 \ m/s$.
$(e)$ Particle velocity amplitude $v_{\max} = a\omega = 5 \times 100\pi = 500\pi \ m/s$.