A transverse sinusoidal wave moves along a string in the positive x-direction at a speed of 10 cm/s. The wavelength of the wave is 0.5 m and its amplitude is 10 cm. At a particular time t, the snap-shot of the wave is shown in the figure. The velocity of point P when its displacement is 5 cm is

Particle velocity v_p is related to the displacement of the particle from the mean position as

$$\begin{gathered} {\mathrm{v}}_{\mathrm{p}}=2\mathrm{\pi v}\sqrt{{\mathrm{A}}^2-{\mathrm{y}}^2} \\ {\mathrm{v}}_{\mathrm{p}}=2\mathrm{\pi }\left(\frac{\mathrm{v}}{\mathrm{\lambda }}\right)\sqrt{{\mathrm{A}}^2-{\mathrm{y}}^2} \end{gathered}$$

$=\frac{2\mathrm{\pi }}{0.5}\times 0.1\sqrt{(0.1{)}^2-(0.05{)}^2}=\frac{\sqrt{3}\mathrm{\pi }}{50} \hat{\mathrm{j}} \mathrm{m}/\mathrm{s}$

Since the wave is sinusoidal moving in positive x-axis the point will move parallel to y-axis. As the wave moves forward in positive X-direction, the point should move upwards i.e., in the positive Y-direction.