# Homework Help: Using Python to calculate projectile motion with resistance

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1. Dec 16, 2015

### Ryaners

I'm trying to write a program to calculate the trajectory of a projectile with air resistance. I've made the following function which calculates the position of the particle based on a given launch angle and initial velocity:

Code (Python):
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
For some reason I'm getting a traceback whenever the angle 'fed' to the function is greater than 1 radian. If I restrict the input angles to anything smaller than π/3, it works. It doesn't for π/2. I can't figure out why this is happening; I've written an almost-identical program to calculate trajectory without air resistance & π/2 isn't a problem for that. Can anyone shed a bit of light on this? This is the error I'm getting:

Here is the whole program, if that helps:

Code (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')
I've been tearing my hair out over this for hours. I'm new to programming so this is probably a basic error I don't have the know-how to spot.

Edit: Just to add - the air resistance is height-dependent.

Last edited: Dec 16, 2015
2. Dec 16, 2015

### Fightfish

I ran a quick test on it; I believe the problem is that your integration time is too short - that is, the object has not hit the ground after 10 seconds. That is why the size of your position array is too small. Try increasing the range of your time axis.

3. Dec 16, 2015

### Ryaners

Ahhh amazing, that's done it! Thank you so much! :)