Final answer:
The maximum velocity of the particles due to the plane wave is found by differentiating the wave function with respect to time and locating the peak value, giving a result of 30 m/s.
Step-by-step explanation:
To find the maximum velocity of the particles of the medium due to the plane wave described by y = 3 cos(x/4-10t-\u03c0/2), we look at the wave equation and differentiate it with respect to time to find the velocity of the particles as a function of time.
The wave function can be written as:
y = A cos(kx - \u03c9t + \u03c6)
Where:
- A is the amplitude
- k is the wave number
- \u03c9 is the angular frequency
- \u03c6 is the phase constant
For this wave, A = 3, k = 1/4, \u03c9 = 10, and \u03c6 = -\u03c0/2.
The velocity of the particles in the medium is the time derivative of the displacement:
v = -A\u03c9 sin(kx - \u03c9t + \u03c6)
The maximum velocity is when the sine function equals 1, which is:
Vmax = A\u03c9
Vmax = 3 \u00d7 10
= 30
Therefore, the maximum velocity of the particles of the medium is 30 m/s.