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).

  1.the balanced   chemical equation is as below

cu +   2 ag(no)₃ → cu(no₃)₂   +2 ag

2.the   much silver   produced   is   50.9 g

step 1: find the   moles of cu

moles =   mass÷   molar mass

  from periodic table the molar mass of cu = 63.5 g/mol

moles is therefore = 15.00 g÷ 63.5 g/mol = 0.236   moles

step 2: use the mole ratio to determine the   moles of ag

from equation above   cu: ag   is 1: 2

therefore the moles of ag   =0.236   moles   x 2/1 =0.472   moles

step 3: find the mass   of ag

mass = moles x molar mass

from periodic table   the molar mass of ag =107.87 g/mol

mass   = 0.472   moles x 107.87 g/mol = 50.9 g

3.   the   much   silver   collected worth is   1.797 oz

  convert   50.9 g   to oz

that is   1 gram = 0.0353 oz

          50.9 grams =?   oz

by cross multiplication

={(50.9 grams x 0.0353 oz) / 1   gram}   =1.797 oz

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