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{\Large \bf Laboratory Calculus I: Lab Project\\
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Bouncing Ball}
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{\bf Introduction}: The famous \index{ magician} magician \index{ Michael Moshen} Michael Moshen has an act in
which he stands in a large 
\index{ triangular sculpture} triangular sculpture and bounces balls off the 
sides, keeping them in motion. This is a look at the elementary  mathematics 
of  what he is doing.

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{\bf The Context to be Investigated:} Two plane surfaces, one parallel to the 
ground, meet at an angle $\theta =  30$ degrees. A ball is thrown against the 
upper plane at an angle $\beta = 5 $ degrees with a velocity of 20 feet per 
second. It hits  10 feet from the corner and five feet above the ground. The  
ball bounces elastically, with no spin or air resistance but is acted upon by 
gravity.

Develop a mathematical model for the path the ball follows with and 
without the influence of gravity. First assume that there is no gravity and
calculate the path of the ball as it bounces between the planes. Vary the
initial velocity, angle of entry, and angle made by the two planes. Can the ball be made to bounce any number of times?

Repeat the investigation with the influence of gravity. Compare the paths
of the ball with and without \index{ gravity} gravity for different initial velocities, angles of entry, angles between the planes, etc.

Finally, repeat the investigation for a ball that is not perfectly elastic.
Suppose the ball always rebounds from a surface with the component of its
\index{ velocity} velocity normal to the surface having only 75\% of its magnitude prior to
impact.

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