A damping force affects the vibration of a spring so that the displacement of the spring is given by y = e-4t (cos 2t + 3 sin 2t). Find the average val

Average velocity = dy/dt = 4sin2t (2t+3) - 4cos2t (6t+1) e^-4t (cos 2t + 3 sin 2t).

Explanation:

Velocity is defined as the rate of change in displacement.

Velocity = change in displacement/time taken

Given the displacement of the string as;

y = e^-4t (cos 2t + 3 sin 2t).

To get the average velocity, we will find the derivative of the displacement with respect to time.

Using function of a function to solve this;

Let u = 4t (cos 2t + 3 sin 2t) ... (1)

y = e^-u ... (2)

Differentiating both functions with respect to their variables we have;

dy/du = - e^-u

du/dt is gotten using the product rule to have;

du/dt = 4t (-2sin2t+6cos2t) + 4 (cos2t+3sin2t)

Opening up the bracket we have;

du/dt = - 8tsin2t+24tcos2t+4cos2t+12sin2t

Collecting like terms;

-8tsin2t+12sin2t+24tcos2t+4cos2t

du/dt = - 4sin2t (2t-3) + 4cos2t (6t+1)

dy/dt = dy/du * du/dt

dy/dt = - e^-u * - 4sin2t (2t+3) + 4cos2t (6t+1)

Substituting u = 4t (cos 2t + 3 sin 2t) into dy/dt, we will have;

dy/dt = - e^-4t (cos 2t + 3 sin 2t) * - 4sin2t (2t+3) + 4cos2t (6t+1)

dy/dt = 4sin2t (2t+3) - 4cos2t (6t+1) e^-4t (cos 2t + 3 sin 2t).

The average velocity of the wave function therefore give us;

dy/dt = 4sin2t (2t+3) - 4cos2t (6t+1) e^-4t (cos 2t + 3 sin 2t).