Abstract:

            This study uses a Java applet to analyze the complex chaotic behavior that occurs in simple mechanical system:  two balls bouncing on top of each other under the influence of gravity.  An elastic collision occurs when a ball collides with the other ball or the ground.  Both energy and momentum are conserved during this process.  When a ball collides with a stationary surface, the velocity of the ball reverses itself.  When a ball collides with a moving surface, the resultant velocities of the two objects do not reverse themselves, but change following the laws of conservation and momentum.  The final velocities differ depending on the mass ratio of the two objects.  In this study, two balls are moving in one-dimensional space allowing both types of collision occur.  If the mass ratio of the top ball to the bottom ball is less than unity, the motion is chaotic almost everywhere.  If the mass ratio is greater than unity, the motion shows a parabolic behavior with chaotic features.  By means of animation and graphical demonstration, the trajectories of the balls can be observed.

Math:          

        After each time step taken by the computer, the states of the balls change in three ways.  The equations for the state change are as follows:

                        t = t + dt;

                        x = x(initial) + v(initial)*t – ½ g*t^2;

                        v = x + v(initial)*t;

The gravity of this system has been set to 1. 

             When the two balls collide, the final velocities of the collision are calculated through the laws of conservation of energy and momentum.  Throughout the entire applet, the time till the two balls collide has been calculated.  When the collision time is less than the next time step, the applet advances the balls forward to the collision time, calculates the change of velocities due to the collision, and then advances the balls the rest of the time step to keep the time step uniform.  The equations for the collision are:

                        center of mass =

                        v = 2*(center of mass) – v;

                        v2 = 2*(center of mass) – v2;

The center of mass is calculated using the mass ratio that the user may change throughout the application.  The mass of the bottom ball is equal to the ratio whereas the mass of the top ball is equal to one minus the ratio.

             When the bottom ball collides with the floor, a simple elastic collision takes place.  This collision time as well is always being calculated throughout the applet.  When this time is less then the next time step, the applet calculates what would occur mathematically when the ball hit the floor.  Since the floor is not moving, the velocity of the ball completely reverses itself.  This is calculated by:

Graph:

            Depending on the mass ratio, the graph can have many different shapes and formations.  When the mass ratio is above 0.5, the formation is completely chaotic.  The overall shape is parabolic, but the points are chaotically interspersed within the boundary.  When the mass ratio is 0.5, or unity, the formation is completely parabolic.  Depending on what the initial values are, the position and velocity can have a limited number of positions that alternate between two similar parabolas.  When the mass ratio is less than 0.5, the formation has both chaotic and uniform sections.  If the position and velocity are near the boundaries of the curve, or if the position is close to zero, then the plot is chaotic.  The center of the graph is not chaotic.  Depending on the position and velocity, the points remain in a circular motion that repeats itself.

            There are two different graphs that the user can decide between.  Both are Poincare Plots, but one of them is the Ball-Ball-Collision Poincare Plot.  In the standard Poincare Plot, the position of the top ball versus the velocity of the top ball is plotted every time the bottom ball collides with the floor.  In the Ball-Ball-Collision Poincare Plot, the position of the top ball versus the velocity of the top ball is plotted every time the two balls collide.

            The Poincare Plot has a feature that the bbc Poincare Plot does not have.  The Poincare Plot allows the user to click an area on the graph to see what the motion and plot of that area will look like.  The bbc Poincare Plot does not have the clickable feature. 

            The plots of the two graphs are very similar.  The difference between the two is that the bbc Poincare Plot seems to be falling to one side.  This is due to the fact that in order to calculate the final positions and velocities, the applet goes through a calculation that is dependent on the mass ratios.  When the ball bounces with the floor, as in the Poincare Plot, the velocity completely reverses itself.  When the balls collide with each other, the final velocites are calculated using the conservation of energy and momentum.  Depending on the mass ratio, the velocities will tend to lean towards one side of the spectrum.  This is the reason the graph looks to be leaning.

References:

   Whelan, N.D., D.A. Goodings, and J.K. Cannizzo.  Two Balls in One Dimension with Gravity.  Physical Review.