# projectile motion with air resistance in python code example

## Example: projectile motion with air resistance in python

# -*- coding: utf-8 -*-

import matplotlib.pyplot as plt
import numpy as np
import math
import scipy.constants as const

g = const.g     #gravitation constant
dt = 1e-3       #integration time step (delta t)
v0 = 40         #initial speed at t=0

angle = math.pi / 4     #launch angle in radians
time = np.arange(0,10, dt)      #create time axis

gamm = 0.005    #gamma (used to compute f, below)
h = 100         #height (used to compute f, below)

def traj_fr(angle, v0):             #function that computes trajectory for some launch angle & velocity
vx0 = math.cos(angle)*v0        #compute x components of starting velocity
vy0 = math.sin(angle)*v0        #compute y components of starting velocity
x = np.zeros(len(time))         #initialise x array
y = np.zeros(len(time))         #initialise y array

x[0],y[0] = 0,0     #initial position at t=0s, ie motion starts at (0,0)
x[1],y[1] = x[0] + vx0*(2*dt), y[0]+vy0*(2*dt)  #calculating 2nd elements of x & y based on init velocity

i=1
while y[i]>=0:  #loop continuous until y becomes <0, ie projectile hits ground
f = 0.5 * gamm * (h - y[i]) * dt        #intermediate 'function'; used in calculating x & y vals below
x[i+1] = ((2*x[i]-x[i-1]) + (f * x[i-1])) / (1 + f)                 #numerical integration to find x[i+1]...
y[i+1] = ((2*y[i]-y[i-1]) + (f * y[i-1]) - g*(dt**2) ) / (1 + f)      # ...& y[i+1]
i = i+1     #increment i for next iteration

x = x[0:i+1]        #truncate x array
y = y[0:i+1]        #truncate y array
return x, y, (dt*i), x[i]       #return x, y, flight time, range of projectile

x,y,duration,distance = traj_fr(angle,v0)   #define variables for output of traj_fr function

print 'Distance: ' , distance
print 'Duration: ' , duration

n = 5
angles = np.linspace(0, math.pi/2, n)   #generate array of n angles
print 'Angles: ' , angles
maxrange = np.zeros(n)                  #generate array of n elements to take range from traj_fr

for i in range(n):      #loop to run angles through traj_fr function & populate maxrange array with distance results
x,y,duration,maxrange[i] = traj_fr(angles[i], v0)

angles = angles / 2 / math.pi * 360       #convert radians to degrees
print 'Launch Angles: ', angles

print 'Optimum Angle: ', angles[np.where(maxrange==np.max(maxrange))]

plt.plot(x,y)       #quick plot of x vs y to check trajectory
plt.xlabel('x')
plt.ylabel('y')