Question

(1 point) A ping-pong ball is caught in a vertical plexiglass column in which the air flow alternates sinusoidally with upward flow at the rate of 3.3m/s and at t = 30 seconds the flow has a minimum (upward) flow of rate of -5.9m/s. (To make this clear: a flow of -5m/s upward is the same as a flow downward of 5m/s. period of 60 seconds. The air flow starts with a maximum 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 through the air. ok times the apparent velocity of the ball the answer should come out about the same -right? What is the average velocity of the air flow? You can average the velocity over one period or over a very long time .(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.) (t) Use the method of undetermined coefficients to find one periodic solution to this equation 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) 2.3. w(t)

Answer 1

The average velocity of the air flow implies [3.3 + (- 5.9)]/2 implies - 1.3 m/s the formula for the velocity of the air flow as a function of time is A [t] = v_avg + A cos [a t] net force (in upward direction) implies mass Times acceleration (in upward direction) - m Times g + k Times (v - v_air) = m Times d v/d t implies d v/d t = [- m Times g + k Times (v - v_air)]/m implies d v/d t = [k Times (v - v_air)]/m - g implies v' (t0 = [k Times (v - v_air)] t/m - g t