from numpy import sqrt,arccos,pi
import numpy as np

'''
Program carousel-range.py

At which angle of deflection should a physical 
pendulum released at the top be released/broken 
to achieve maximum horizontal distance from the 
pendulum's axis? How far will it fly (w/o friction)?

x  = max range in units of radius R of the pendulum
co = cos(theta)
'''

x = 0.
N = 1000000
dc = 0.05/N
for i in range(N):
    co = 0.75 - dc*i
    xprev = x
# calculate the range x          
    si = sqrt(1.-co*co)  # sin(theta)
    tmp = 1.+co   
    x = si +2.*co*(tmp*si + sqrt(tmp*(2.-co*co*tmp)) ) 
# print(i,rangex(c))
    if (xprev > x): break
        
co = co + dc/2
print('\n max found @step', i, '\n',
      'max range/R =',np.around((x+xprev)/2,8), '\n',
      'cos(theta),  theta[deg]:\n', np.around(co,9),' ', np.around(arccos(co)*180/pi,8), '\n')