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### Motion Lab

You've done a lot of problems so far where we have an equation and you decide what the solution is. In this lab we are going to reverse the process. Below are the graphs of 7 different solutions to equations of the form $mx''+cx'+kx=F_0 \cos(\omega t)$, $x(0)=x_0$, $x'(0)=x_1$. You are to find the initial value problems for which they are solutions. To help, we've got an applet that will let you input an initial value problem and graph the solution. However, don't just randomly type in coefficients. Looking at the graphs of the solutions, you should be able to decide what the functions are (well, except for the transient part of one problem). Then think about what sort of problem has that solution. Note that this is actually closer to actual engineering practice than just solving equations. You have a behavior you want to get and have to design the circuit to give it to you. While mathematically you can have either positive or negative coefficients for an equation, in practical problems the coefficients are usually all positive (since they have real meaning like mass or coefficient or friction and those are positive). In the graphs below, the coefficients of the equation, though perhaps not the initial values, are all positive.

Launch the motion lab. Change coefficients as you want, then click the "Update Graph" button to see the graph of the solution. The "Reset" button will return the coefficients to the last set of values that you graphed. This means that while you are adjusting coefficients the equation and the graph will temporarily be out of sync, but pressing either button will bring them back into sync, either by making the graph match the equation (Update Graph) or resetting the equation to match the graph (Reset). Note that some problems have different graph windows and you should update the graph window on the applet to agree with the picture you are trying to match.

1. Can you find an unforced problem where the solution curve never crosses the axis?
Can you find an unforced problem where the solution curve crosses the axis exactly once?
Can you find an unforced problem where the solution curve crosses the axis exactly twice?
Can you find an unforced problem where the solution curve crosses the axis infinitely many times?