Ball speed

From cannon to gutter, assume the complete 
travel path of the ball can be broken into 
two paths.  

The first path is out from the cannon and all 
along the outfield rail, in the ball channel.  

The second path is the curve the ball takes 
after friction has slowed its velocity to the 
point where gravity pulls it onto the track.

Vb = R * ThetaDotb

Assume V0 = R * Theta0 is the initial velocity 
of the ball coming out of the cannon.

For the first path, we can either 

 (1) assume friction independent of the speed of 
 the ball.  For the ball channel assume f = Fbc, 
 or in the first path d2Theta/dt2 = -Fbc.

Integrate:

 ThetaDotb = -Fbc * t + ThetaDot(0)

 a straight linear decrease.

 (2) assume friction proportional to normal force
 proportional to (Vb^2/R).

 d ThetaDotb / dt = - k*R*ThetaDotb^2

 dv/dt = -k v^2
 v = -k v^3 + ThetaDot(0)

 a much quicker decrease.


CRITICAL SPEED
