The velocity of a particle moving with simple harmonic motion is maximum at the mean position.

Explanation:
Simple harmonic motion (SHM) is a type of periodic motion in which the restoring force is directly proportional to the displacement of the particle from the equilibrium position and is always directed towards the equilibrium position. The motion of a particle undergoing SHM can be described by the equation x = A sin(ωt + φ), where x is the displacement of the particle from the mean position, A is the amplitude of the motion, ω is the angular frequency, t is the time, and φ is the phase constant.

The velocity of the particle is given by the derivative of the displacement equation with respect to time, v = dx/dt = Aω cos(ωt + φ). At the mean position, the displacement of the particle is zero, and therefore the cosine term becomes cos(φ). The maximum value of the cosine function is 1, so the maximum velocity occurs when cos(φ) = 1. This means that the phase constant φ must be equal to 0 or a multiple of 2π.

At the mean position, the particle is momentarily at rest and starts to move in the opposite direction. As it moves away from the mean position, the velocity increases until it reaches a maximum value. The particle then slows down, comes to a momentary rest at the extreme position, and starts to move towards the mean position again. The velocity continues to increase until it reaches another maximum value at the mean position. Therefore, the velocity of the particle is maximum at the mean position in SHM.

In summary, the velocity of a particle moving with simple harmonic motion is maximum at the mean position because the particle changes its direction of motion and starts to move in the opposite direction at this point.