Worksheet #7 - PHY102 (Spr. 2005)

Collisions

Due Thursday March 3rd

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*)

f1[x_,y_]:=a*x+b*y-c

f2[x_,y_]:=c*x+d*y-e

sol=Solve[{f1[x,y]==0,f2[x,y]==0},{x,y}]

{x,y}={x,y}/. sol[[1]]

(*This checks to see that the solutions are correct*)

Simplify[f1[x,y]]

Simplify[f2[x,y]]

Problem 1

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:
(i) m=M
(ii) m=2M

Problem 2

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?