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differential equation applications 225Here atan2 is a function in most computer languages that computes the arctangentin the correct quadrant without requiring any explicit divisions (that canblow up).9.14.3 Assessment1. Apply the rk4 method to solve the simultaneous second-order ODEs (9.62)and (9.63) with a 4-D force function.2. The initial conditions are (a) an incident particle with only a y component ofvelocity and (b) an impact parameter b (the initial x value). You do not needto vary the initial y, but it should be large enough such that PE/KE ≤ 10 −10 ,which means that the KE ≃ E.3. Good parameters are m =0.5, v y (0)=0.5, v x (0)=0.0, ∆b =0.05, −1 ≤ b ≤ 1.You may want to lower the energy and use a finer step size once you havefound regions of rapid variation.4. Plot a number of trajectories [x(t),y(t)] that show usual and unusual behaviors.In particular, plot those for which backward scattering occurs, andconsequently for which there is much multiple scattering.5. Plot a number of phase space trajectories [x(t), ẋ(t)] and [y(t), ẏ(t)]. How dothese differ from those of bound states?6. Determine the scattering angle θ = atan2(Vx,Vy) by determining the velocityof the scattered particle after it has left the interaction region, that is,PE/KE ≤ 10 −10 .7. Identify which characteristics of a trajectory lead to discontinuities in dθ/dband thus σ(θ).8. Run the simulations for both attractive and repulsive potentials and for arange of energies less than and greater than V max = exp(−2).9. Time delay: Another way to find unusual behavior in scattering is to computethe time delay T (b) as a function of the impact parameter b. The time delayis the increase in the time it takes a particle to travel through the interactionregion after the interaction is turned on. Look for highly oscillatory regionsin the semilog plot of T (b), and once you find some, repeat the simulationat a finer scale by setting b ≃ b/10 (the structures are fractals, see Chapter 13,“Fractals & Statistical Growth”).9.15 Problem 2: Balls Falling Out of the SkyGolf and baseball players claim that hit balls appear to fall straight down out of thesky at the end of their trajectories (the solid curve in Figure 9.9). Your problem is todetermine whether there is a simple physics explanation for this effect or whetherit is “all in the mind’s eye.” And while you are wondering why things fall out ofthe sky, see if you can use your new-found numerical tools to explain why planetsdo not fall out of the sky.−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 225