## Falling p.1

The above diagram plots various the energies involved
as a function of height above the ground (*z*). The
line representing "Total Energy" is flat, because total energy is
conserved (i.e., is constant). Potential energy
(*U(z)*) is plotted in red: for *z*>0,
*U(z)*=*mgz*, for *z*<0 *U(z)*
is infinite (the bouncing ball cannot penetrate the ground).
Total energy is the sum of
kinetic energy and potential energy: thus kinetic energy
is what you have to add to potential energy to get the
constant total energy. Kinetic energy must be positive,
so the only places the particle can actually go are where
*U(z)*<*E*. The points on the edge of
the allowed region (where *U(z)*=*E*, i.e.,
*z*=0 and *z*_{max}) are called turning
points. There the kinetic energy is zero, and so the velocity
of the particle is zero. Just prior to reaching *z*_{max},
the particle is moving up (positive velocity). Just after reaching
*z*_{max} the particle is moving down (negative
velocity). At *z*_{max} the particle is momentarily at rest.
The below plots the position and velocity as functions of time.

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