A ping-pong ball is caught in a vertical plexiglass column in which the air flow

Question

A ping-pong ball is caught in a vertical plexiglass column in which the air flow alternates sinusoidally with a period of 60 seconds. The air flow starts with a maximum upward flow at the rate of 4m/s and at t=30 seconds the flow has a minimum (upward) flow of rate of ?6m/s. (To make this clear: a flow of ?5m/s upward is the same as a flow downward of 5m/s. The ping-pong ball is subjected to the forces of gravity (?mg) where g=9.8m/s2 and forces due to air resistance which are equal to k times the apparent velocity of the ball through the air. What is the average velocity of the air flow? You can average the velocity over one period or over a very long time -- the answer should come out about the same -- right?__________________________(Include units.) Write a formula for the velocity of the air flow as a function of time. A(t)=__________________________ Write the differential equation satisfied by the velocity of the ping-pong ball (relative to the fixed frame of the plexiglass tube.) The formulas should not have units entered, but use units to trouble shoot your answers. Your answer can include the parameters m - the mass of the ball and k the coefficient of air resistance, as well as time t and the velocity of the ball v. (Use just v, not v(t) the latter confuses the computer.) v?(t)= _________________________ Use the method of undetermined coefficients to find one periodic solution to this equation: v(t) = __________________________ Find the amplitude and phase shift of this solution. You do not need to enter units. v(t)=___________________________ Find the general solution, by adding on a solution to the homogeneous equation. Notice that all of these solutions tend towards the periodically oscillating solution. This is a generalization of the notion of stability that we found in autonomous differential equations. Calculate the specific solution that has initial conditions t=0 and w(0)=4.2. w(t)=___________________________

Explanation / Answer

. What is the average velocity of the air flow? Vavg = (-.4 + (-.4))/2 = -.4 m/s B. Write a formula for the velocity of the air flow as a function of time. A(t) = A + Bsin((pi)t/30) then use initial conditions of A(0) = -.4 and A(30) = -.4 to solve for the coefficients of A and B.----------------

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