Velocity of a Plane Progressive Wave

The given equation of the plane progressive wave is: $y = 5 \cos \pi \left(200 t - \frac{x}{150}\right)$

Expanding the argument inside the cosine function, we get: $y = 5 \cos \left(200\pi t - \frac{\pi x}{150}\right)$

Comparing this with the standard form of a progressive wave equation: $y = A \cos(\omega t - kx)$

we identify the angular frequency ($\omega$) and the propagation constant ($k$) as: $\omega = 200\pi \text{ rad/s}$ $k = \frac{\pi}{150} \text{ rad/cm}$

The velocity of the wave ($v$) is given by the relation: $v = \frac{\omega}{k}$

Substituting the values of $\omega$ and $k$: $v = \frac{200\pi}{\frac{\pi}{150}} = 200 \times 150 = 30000 \text{ cm/s}$

Converting the wave velocity from $\text{cm/s}$ to $\text{m/s}$: $v = \frac{30000}{100} \text{ m/s} = 300 \text{ m/s}$

Hence, the velocity of the wave is $300 \text{ m/s}$, which corresponds to option D.