Worksheet #5 - PHY102 (Spr. 2007)Solving Equations and Differential Equations IIDue Thursday February 15thLast week you learned about the following functions which are among themost important in Mathematica:Solve - solves polynomial equations analyticallyNSolve - solves polynomial equations numericallyFindRoot - solves all equations numericallyDSolve - 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 that by hand. This worksheet is intended tohelp you learn some more about setting up and solving physics problems.Problem 1.(i) A ball is falling vertically through a fluid. Apart from gravity, a dragforce Fdacts on the ball. The drag force opposes the motion and increasesin proportion to the velocity (~Fd= −k~v, where k is a drag coefficient thatdepends on the fluid). Find and plot the time dependence of the position andvelocity of a 100g ball which is released from rest at t = 0, in a fluid withdrag coefficient k = 0.02. Choose a time range which shows the terminalvelocity of the ball. Note that it is easiest to solve this problem by solvingfor the motion analytically using DSolve.(ii) Consider a cannon on a 500m high hill. Assuming that the cannonfires 0.1kg cannonballs horizontally with initial velocity 500m/s, find therange of the cannon for a drag coefficient k = 0.01. How long does it takethe cannonball to reach ground?Problem 2.1You lift a box of mass 30kg vertically a height of h meters. However youdecide that you want your little brother to lift the other 30 boxes that mustbe lifted to the same height. Since he is weaker than you, you kindly put aninclined plane (at angle θ to the horizontal) in place to assist him.(i) If there is no friction, use mathematica to evaluate the integralR~F .~drand hence prove that you and he do the same amount of work per box.(ii) Now consider adding friction f = µxN which increases as the boxmoves up the plane. Here x is distance along the inclined plane, µ is thefriction coefficient and N is the force normal to the inclined plane. Howmuch additional work does your brother do (per box) due to the friction onthe inclined plane? (Solve this problem analytically using
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