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(C) In a sine wave,the displacement of a particle is given by $y = A \sin(\omega t - kx)$.The velocity of a particle is $v_p = \frac{dy}{dt} = A\omega

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$\frac{\lambda}{2}$

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(C) In a sine wave,the displacement of a particle is given by $y = A \sin(\omega t - kx)$.The velocity of a particle is $v_p = \frac{dy}{dt} = A\omega \cos(\omega t - kx)$.For two particles at positions $x_1$ and $x_2$ to have the same speed,their velocities must satisfy $|v_{p1}| = |v_{p2}|$.This occurs when the phase difference is an integer multiple of $\pi$. Specifically,particles separated by a distance $\Delta x = \frac{\lambda}{2}$ have velocities that are equal in magnitude but opposite in direction $(v_1 = -v_2)$,meaning their speeds are identical.Thus,the minimum distance between two particles having the same speed is $\frac{\lambda}{2}$.

$A$ sine wave of wavelength $\lambda$ is travelling in a medium. The minimum distance between two particles,which always have the same speed,is

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