Worksheet #5 – PHY102 (Spring 2012)Solving Equations and Differential Equations IILast week you learned the following functions, which are among the most important in Mathemat-ica:Solve – solves equations analyticallyNSolve – solves polynomial equations numericallyFindRoot – solves equations numerically (Requires a starting point)DSolve – solves differential equations analyticallyNDSolve – solves differential equations numericallyOnce you know what these routines do and how to use them, you have a very powerful set of toolsfor solving problems in physics. However the hardest part of physics is to set up the mathematicaldescription of the problem, and that you still need to do by hand. This worksheet is intended t ohelp you learn some more about setting up and solving physics problems.Problem 1.First review the various method to solve equations:(i) Use Plot[{Exp[Sqrt[x]], 2*x}, {x, ??, ??}] with appropriate choices of the limits toroughly estimate the values of x for which Exp[Sqrt[x]] = 2 x.(ii) Use NSolve to find numerical values of the solutions.(iii) Use Solve followed by using N to find the same numerical values.(iv) Use FindRoot to again find the numerical values of the solutions.Problem 2.A ball is falling vertically through a flu id. In addition to gravity (use g = 9.80 m/s2), a drag forceFdacts on the ball. The drag force opposes the motion and increases in proportion to the velocity:~Fd= −k~v, where k is a drag coefficient that depends on the fluid. (This may or may not bean accurate physics approximation—that depends on the how big th e velocity is, and for typicalapplications involving air resistance, a force proportional to v2would be much closer to the truth.But it is a very convenient approximation, because it simplifies the mathematics by decoupling theequations of motion in the x and y directions.)(i) Find and plot the time dependence of the position and velocity of a 100 g ball that is releasedfrom rest at t = 0 in a fluid with drag coefficient k = 0.02. Choose a time range that shows theapproach to terminal velocity of the ball. Hint: it is easiest to solve this problem by solving forthe motion analytically using DSolve.1(ii) Find the terminal velocity to an accuracy of 1 part in 105.Problem 3.Consider a cannon at the top of a 500 m high hill. Assume that the cannon fires 0.1 kg cannonballshorizontally with initial velocity 500m/s, and k = 0.01 .(i) How long does it take the cannonball to reach the ground?(ii) Find the range (= horizontal distance traveled).(iii) If the cannon is fired at an angle θ above the horizontal, what angle gives the maximum r ange,and what is that
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