Worksheet #7 - PHY102 (Spr. 2006)CollisionsDue Thursday 9pm March 2th, 2006In this worksheet, we will return to solving equations and solving differentialequations.Often there are multiple ways of accomplishing something in Mathematica.Usually one way is easier than another but less elegant. Why might you wantto use the elegant method rather than the “easy” one? Because it can oftensave trouble later on in your mathematica session. Here is an example. Let’ssay you want to know the distance a mass of 50kg falls in 30s after falling outof an airplane. Obviously we need to use y = v0t+12at2where v0= 0, t = 30sand a = −g = −9.81m/s2. The simplest way is to type into mathematica:y = -9.81*30ˆ2/2.An obviously more elegant route is to type:v0=0;a=-9.81;t=30;y=v0*t + a*tˆ2/2or a more space-saving way would be:{v0,a,t}={0,-9.81,30};y=v0*t + a*tˆ2/2The problem with both of these approaches may come later because you havepermanently defined the variables v0, a and t to these values and whereverthey appear later in your Mathematica notebook these values will be sub-stituted leading to unpredictable results. (It is to clean up messes like thisthat we regularly use Remove[“Global‘*”]). The most elegant solution tothis problem is to define the equation as an equation then seek the solutionwith particular “substitutions”:y[t ] := v0*t + a*tˆ2/2sol1 = y[t] /. {v0 → 0, a → -9.81, t → 30}This solution is elegant because the function of interest is defined as a func-tion so we can operate on it (for example, find its derivative etc.). We foundthe particular solution we were looking for (i.e., got the same result as wedid when we used the “easy” methods shown above) but didn’t permanentlyreset the values of any internal variables in mathematica. Try it. Enter the1different commands above. Then check what Mathematica thinks the vari-ables (e.g., a and v0) are after each case. Use a Remove[“Global‘*”] inbetween each test. Many of you have already been using this “substitution”technique if you have been using the DSolve example given out with work-sheet 4. When you use DSolve or Solve, for example, the solutions to theequation are returned as a list of substitutions. This is discussed in moredetail below.Enter the following code:(*You can structure your equation solvers like this*)(*This solves two simultaneous linear equations*)f1[x ,y ]:=a*x+b*y-cf2[x ,y ]:=c*x+d*y-esol=Solve[{f1[x,y]==0,f2[x,y]==0},{x,y}]{x,y}={x,y}/. sol[[1]](*This checks to see that the solutions are correct*)Simplify[f1[x,y]]Simplify[f2[x,y]]Problem 1Use mathematica to solve the following problem (see the example above)A ball of mass m moving horizontally with a velocity u undergoes a head onelastic collision with another ball of mass M travelling at velocity U. Applyconservation of momentum and energy to find expressions for the final ve-locities of these two particles as a function of m, M, u and U. Verify yoursolutions by confirming that they preserve energy and momentum conserva-tion.Now find the final velocities for the following cases:(i) m = M(ii) m = 2MProblem 2A particle of mass m strikes a pendulum of length l and of mass M whichis initially at rest and becomes embedded in it. The center of mass of thependulum(plus particle) rises a vertcial distance h.(i) Find the initial speed of the particle. Find the maximum angle bywhich the pendulum swings.2(ii) Now suppose m = 1, M = 9, g = 10.0m/s2, h = 4m, l = 10m.Solve the non-linear differential equation for the pendulum’s motion. Henceplot the behavior of the pendulum angle as a function of time. Now solvethe linear pendulum problem using the same parameters. Plot the timedependent oscillations of the linear and non-linear solutions on the samegraph. Do you think it is legitimate to approximate the motion in thisproblem by the linear(simple) pendulum
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