Collisions

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*=2*M*

**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*/*s*^{2} , *h* = 4*m*, *l*=10*m*. 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?