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2.5. Unsolved Problems Exercise 2.1: Exploding projectiles A projectile is launched at 20 m/s at an angle of 30 with the horizontal. In the course of its flight, it explodes, breaking into two fragments, one of which has twice the mass of the other. The two fragments land simultaneously. The lighter one lands 20 m from the launch point in the direction in which the projectile was fired. Part (a) Where does the other fragment land? Part (b) Plot the motion of the two fragments and the motion of the center of mass point. Part (c) Write a function that will plot the motion of the two fragments and the motion of the center of mass when the initial velocity, mass of fragments, and position of one of the fragments is given. Exercise 2.2: Intersecting trajectories A hunter with a blowgun wishes to shoot a monkey hanging from a branch. The hunter aims right at the monkey and fires the dart. At the same instant, the monkey lets go of the branch and drops to the ground. Part (a) Show that the monkey will be hit regardless of the angle of the dart to the monkey and the dart's initial velocity so long as it is great enough to travel the horizontal distance to the tree before the dart hits the ground. Part (b) Plot the points for the positions of the dart and the monkey when those positions are separated by equal time intervals. Choose the time intervals so that the last point is the intersection of the dart and monkey. Part (c) Do the plot for several different initial values of the dart and for several different heights of the monkey. Exercise 2.3: Reflecting trajectories A boy stands a distance x0 from a vertical wall and throws a ball. The ball leaves the boy's hand at a height y0 above the ground with initial velocity v0. When the ball hits the wall, its horizontal component of velocity is reversed and its vertical component remains unchanged. Part (a) Where does the ball hit the ground? Part (b) Plot the motion of the ball for several different initial velocities. Exercise 2.4: Principle of Galilean invariance Two balls are released simultaneously from a platform above Earth's surface. One is merely dropped from rest, while the other is launched with an initial horizontal velocity v0. Graph the stroboscopic images of the two balls at successive instants of time and show that they fall downward in unison, reaching equal heights at the same time. Consider several different values of v0. Exercise 2.5: Elastic collision A ball moving with nonrelativistic velocity vi makes an off-center, perfectly elastic collision with another ball of equal mass initially at rest. The incoming ball is deflected at an angle from its original direction of motion. Part (a) Find the velocities and trajectories of the balls after the collision. Part (b) Plot the trajectories and center of mass point for different values of vi and . Exercise 2.6: Quadrupole Consider three charges located on the z-axis. One negative charge of magnitude q is located at +s, the other, of the same magnitude, is at s. A positive charge of magnitude 2q is located at the origin. The field point is at r, and it is assumed that the separation s of the charges is small compared to the distance r. Part (a) Find the potential and electric fields to leading order in s/r. Part (b) Plot the equipotential surfaces and the electric field lines. Exercise 2.7: Fast Fourier transform Consider the function 1 1 Sin 2 t 1 Sin 6 t Sin 8 t 10 5 Generate a table of data points from this function with random noise added. Take the Fourier transform of the table and plot the results. Exercise 2.8: B field of a spinning disk A thin disk of charge density s, radius R, and thickness t R rotates with an angular velocity about the z-axis. Find the B field on the axis of the uniformly rotating disk and plot the results. Exercise 2.9: Fit to data Consider the following data table: In[l]: data Table[{ x, 5 Sin[3 Pi x] + Random[ Real, {-.2, .2} ] } // N //Chop[#,10^-3]&, {x, 0, 0.5 ,0.05} ]; Part (a) Fit the data to a power series in x and plot the fit and data on the same graph. Part (b) Use leastSquares to fit the data to A Sin[B x] and plot the fit and data on the same graph.