Worksheet #5 – PHY102 (Spring 2010)Solving Equations and Differential Equations IILast week you learned the following functions, which are among the mostimportant in Mathematica: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 avery powerful set of tools for solving problems in physics. However the hard-est part of physics is to set up the mathematical description of the problem,and that you still need to do by hand. This worksheet is intended to helpyou 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 choicesof the limits to roughly 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 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 fluid. In addition to gravity (useg = 9.80 m/s2), a drag force Fdacts on the ball. The drag force opposes themotion and increases in proportion to the velocity:~Fd= −k~v, where k isa drag coefficient that depends on the fluid. (This may or may not be anaccurate physics approximation—that depends on the how big the velocity is,and for typical applications involving air resistance, a force proportional to v2would be much closer to the truth. But it is a very convenient approximation,1because it simplifies the mathematics by decoupling the equations of motionin the x and y directions.)(i) Find and plot the time dependence of the position and velocity of a100 g ball that is released from rest at t = 0 in a fluid with drag coefficientk = 0.02. Choose a time range that shows the approach to terminal velocityof the ball. Hint: it is easiest to solve this problem by solving for the motionanalytically using DSolve.(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 thecannon fires 0.1 kg cannonballs horizontally with initial velocity 500m/s, andk = 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 anglegives the maximum range, and what is that
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