Answer:
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).
