Final answer:
To find the speed of a particle in a simple harmonic traveling wave at a specific location and time, differentiate the given wave equation with respect to time and evaluate it using the provided values for location and time.
Step-by-step explanation:
To calculate the speed of a particle at a specific location and time for a simple harmonic traveling wave, we use the provided wave equation y(x, t) to derive the velocity. The given wave equation is y(x, t) = (2/1) sin 1(3x + 2t), with x and y in centimeters and t in seconds. The particle's velocity v(x, t) can be found by differentiating the displacement equation with respect to time t.
The general equation for the velocity in simple harmonic motion (SHM) is v(t) = - Umax sin T', where Umax is the maximum speed of the particle and is given by Umax = √(k/m)X. However, in the given wave equation parameters such as spring constant k and mass m are not provided, therefore, we directly differentiate y(x, t) with respect to t to find v(x, t).
Given the wave equation y(x, t), the velocity equation will be v(x, t) = d/dt [ (2/1) sin(3x + 2t) ]. Using the chain rule, the differentiation of the sine function with respect to time t yields v(x, t) = 2 cos(3x + 2t) * 2, since the derivative of sin with respect to its argument is cos, and we multiply by the derivative of the argument (3x + 2t) with respect to t, which is 2.
The next step is to plug in the values x = 4 cm and t = 2 s to find v(4 cm, 2 s). So v(4, 2) = 2 cos(3*4 + 2*2) * 2 = 2 cos(12 + 4) * 2 = 2 cos(16) * 2. Calculating cos(16) in radians and multiplying through gives us the velocity at that specific location and time.