Saloon Door Problem: Saloon doors can swing through the door frame. A good door damper will slow a swinging door down so it does not swing through the door frameunless you shove the door hard toward the frame. In that case, it will swing through and return from the other side, which is a characteristic of "overdamping." This problem will study the damping effect, using the Mathlet Damped Vibration. Notation used in this activity: is Newton's notation for derivative with respect to time. v Velocity of the Salon Doors Step #1: Open the applet. The default is about solutions of the second order homogeneous equation +b+kx = 0. The initial conditions are set using the box at left. The horizontal direction gives x(0) and the vertical direction gives (0) . The right graphing window displays the corresponding solution. Step #2: Click on the boxes for "show trajectory" and "relate graphs" then click on the advanced double arrows below the right-side graph. Step #3: Move the cursor around in the initial conditions box. Observe the behavior of the left end of the graph in the right window. Verify for yourself that the slope increases when the horizontal coordinate increases, and that the value increases when the vertical coordinate increases. These coordinates can also be controlled using the sliders along the edge of the initial conditions box. Notice as you decrease the damping constant b, you willsee that the system becomes "underdamped", which means the solutions oscillate. In this problem we will study the case in which b = 2 and k = 0.75. (The applet won't let you choose k = 0.75. Approximate it by 0.74.) Check that this is overdamped...which means the characteristic polynomial has two distinct real roots, and the solutions do not oscillate. Step #4: Setting the conditions Set x(0) = 0.50. Set the (0) slider at 1.00 and start to increase it slowly. Watch the effect on the solution curve. At first it swings through, but soon appears not to swing through. You can get a better picture by zooming in using one of the power-of- ten buttons. Work with this until you come up with the smallest value of (0) which does not result in a solution crossing the horizontal axis. (It will be a negative number.) 1. Give the value youdiscover. 2. Find the general solution of this ODE, with these values of k =0.75 and b =2. 3. Express the constants of integration in terms of (0) using x(0) = 0.5 4. What is the solution with (0) = v? This is the value you found in 1 above. 5. As t approaches infinity, what value does the graph approach? 6. Identify the steady-state and transient solutions.