In this worksheet, we will return to solving equations and solving differential equations. Enter the following code:
(*You can structure your equation solvers like this*)
(*This solves two simultaneous linear equations*)
(*This checks to see that the solutions are correct*)
Use mathematica to solve the following problem (see the example above) A ball of mass m moving horizontally with a velocity u undergoes a head on elastic collision with another ball of mass M travelling at velocity U. Apply conservation of momentum and energy to find expressions for the final velocities of these two particles as a function of m, M, u and U. Verify your solutions by confirming that they preserve energy and momentum conservation.
Now find the final velocities for the following cases:
A particle of mass m strikes a pendulum of length l and of mass M which is initially at rest and becomes embedded in it. The center of mass of the pendulum(plus particle) rises a vertcial distance h.
(i) Find the initial speed of the particle. Find the maximum angle by which the pendulum swings.
(ii) Now suppose m=1, M=9,
g=10.0 m/s2 , h = 4m, l=10m. Solve the non-linear differential
equation for the pendulum's motion. Hence plot the
behavior of the pendulum angle as a function of time.
Now solve the linear pendulum problem using the same
parameters. Plot the time dependent oscillations
of the linear and non-linear solutions on the
same graph. Do you think it is legitimate to approximate
the motion in this problem by the linear(simple) pendulum equation?