Solving Equations and Differential Equations II

Due Thursday 17th February

Due Thursday 17th February

Last week you learned about the following functions which,
for physicists, are among the most useful in Mathematica:

Solve - solves polynomial equations analytically

NSolve - solves polynomial equations numerically

FindRoot - solves all equations numerically

DSolve - solves differential equations analytically

NDSolve - solves differential equations numerically

Once you know what these routines do and how to use them, you
have a very powerful set of tools for solving problems in physics.
However the hardest part of physics is to set up the mathematical
description of the problem, and that you need to practice. This
week we work on setting up mechanics problems and then
solving them using mathematica.

**Problem 1.**

(i) A ball is falling vertically through a fluid. Apart from
gravity, a drag force *F*_{d}
acts on the ball. The drag force
opposes the motion and increases in proportion to the speed
(
, where
*k* is a *drag coefficient* that
depends on the fluid). Find and plot the time
dependence of the position and velocity of
a 100*g* ball which is released from rest at *t*=0, in a fluid
with drag coefficient *k*=0.02. Choose a time range which
shows the terminal velocity of the ball.

(ii) Now consider that the same ball in the same fluid
is given an initial
velocity
*v*_{x} = 10*m*/*s*
in the horizontal direction.
Use mathematica to find the motion in the x and y directions.
Plot the displacement in the x direction as a function of time
and plot the trajectory(parametric plot) of the ball.

(iii) Consider a cannon on a 500*m* hill. Assuming that the
cannon fires 10*kg* cannonballs horizontally with initial
velocity 500*m*/*s*, find the range of the cannon
for a drag coefficient of *k*=.002.
Compare this to the range
in the absence of drag.

**Problem 2.**

You lift a box of mass
m=30*kg*
vertically a height of *h* meters.
However you decide that you want your little brother to lift the other
30 boxes that must be lifted to the same height. Since he is
weaker than you, you kindly put an inclined plane(at angle to the horizontal) in place to assist him.

(i) If there is no friction, use mathematica to evaluate the integral and hence prove that you and he do the same amount of work per box.

(ii) Now consider adding friction
, where
is the dynamics friction coefficient and *N* is the
force normal to the inclined plane.
How much additional work does your brother
do (per box) due to the friction on the inclined plane?
(Solve this problem analytically using Mathematica)