Final answer:
To calculate the vibrational speed of a particle in a wave, we use the derivative of the wave equation with respect to time. By inserting the given values into the relevant wave equations, we can solve for the speed at a specified location and time.
Step-by-step explanation:
The student's question is about calculating the vibrational speed of a particle in a progressive wave. Given that a progressive wave is propagating in the positive x-direction at a velocity of 5 m/s, with an amplitude of 15 mm (0.015 m) and a wavelength of 40 cm (0.4 m), we want to find the speed of a particle at position x = 0.5 cm (0.005 m) at time t = 0.2 s.
The equation describing the wave can be assumed in the form:
y(x, t) = A-*sin(kx - ωt + φ)
Where:
- A is the amplitude
- k is the wave number (k = 2π/λ)
- λ is the wavelength
- ω is the angular frequency (ω = 2πf)
- f is the frequency (f = v/λ)
- φ is the phase constant, which we'll take as zero since we're looking at x = 0 at t = 0
The vibrational speed (v_p) of the particle is the rate of change of y with respect to time (dy/dt).
We can differentiate y with respect to t to find:
v_p = dy/dt = -Aω*cos(kx - ωt)
Substituting the given values into the formulas, we estimate:
k = 2π/0.4 m = 5π m⁻¹
f = 5 m/s / 0.4 m = 12.5 Hz
ω = 2π * 12.5 Hz = 25π s⁻¹
Then, plug in the values to calculate the vibrational speed at the specific position and time:
v_p = -0.015 m * 25π s⁻¹ * cos(5π * 0.005 m - 25π * 0.2 s)
The cosine term can be simplified, and the vibrational speed can be calculated accordingly.