## Falling p.9

### Falling in 2 Dimensions: Ballistic Motion

Consider ballistic or projectile motion: where there is no force
in one direction (*F*_{y}=0) and a constant force
in the other (*F*_{z}=-*mg*). Classically the
motion in the *y* and *z* directions totally separates:
one does not depend on the other.
We have accelerated motion in the *z* direction:

*z(t)* = *z*_{0} + *v*_{z0}*t* - ½*gt*^{2}:

*v*_{z}*(t)* = *v*_{z0} - *gt*

where the initial height
(*z*_{0}) and the initial *z* velocity (*v*_{z0})
are the boundary conditions needed to solve classical equation of motion:

*z''* = -*g*

and uniform (constant velocity) motion in the *y* direction:

*y(t)* = *y*_{0} + *v*_{y0}*t*

*v*_{y}*(t)* = *v*_{y0}

where the initial down-range position
(*y*_{0}) and the initial *y* velocity (*v*_{y0})
are the boundary conditions needed to solve classical equation of motion:

*y''* = 0

Motion in the *y* direction is "free" (i.e., free from forces) and unbounded
(i.e., if *v*_{y0} is not zero, over all time, the particle reaches every
possible *y* position). Motion in the *z* direction is bounded between
the perfectly elastic ground (at *z*=0) and some maximum height
(*z*_{max}) that depends on the initial "*z*" energy
*E*_{z}. (Note that energy is certainly not a vector, rather
the totally separate differential equations for *z* and *y*
have separate constants-of-motion which exactly correspond to the one-dimensional
concepts of energy. Thus:

*E*_{z} = ½*m v*_{z}^{2} + *mgz*

and

*E*_{y} = ½*m v*_{y}^{2}

are separately conserved quantities that sum to form the total energy *E*.)

Classically the trajectory is a series of parabolic bounces (in blue):

Note that the full separation of *y* motion from *z* motion
means that positions that are energetically allowed like:

*E/mg* > *z* > *z*_{max}

cannot actually occur. A slight coupling of *y* and *z*
motion (e.g., that would occur if the *z*=0 ground plane were
slightly bumpy) would expand the allowed region to include the above region.

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