A wave travels along the positive x -direction with a speed of 20 m...

The given wave equation is $y = 0.20 \sin(\pi x - 2\pi \times 10^3 t)$, where $y$ and $x$ are in cm and $t$ is in s.

Substituting $x = 2.0$ cm and $t = 0$ s into the equation:

$y = 0.20 \sin(\pi \times 2.0 - 0) = 0.20 \sin(2\pi) = 0$

The velocity of the particle is given by the partial derivative of displacement with respect to time:

$v_p = \dfrac{\partial y}{\partial t} = \dfrac{\partial}{\partial t} [0.20 \sin(\pi x - 2\pi \times 10^3 t)]$

$v_p = 0.20 \times (-2\pi \times 10^3) \cos(\pi x - 2\pi \times 10^3 t)$

$v_p = -400\pi \cos(\pi x - 2\pi \times 10^3 t) \text{ cm s}^{-1}$

Substituting $x = 2.0$ cm and $t = 0$ s:

$v_p = -400\pi \cos(2\pi) = -400\pi \text{ cm s}^{-1}$

Converting to $\text{m s}^{-1}$:

$v_p = -4\pi \text{ m s}^{-1}$

Thus, the displacement is zero and the velocity is $-4\pi \text{ m s}^{-1}$.