This is a file from the Wikimedia Commons


From Wikipedia, the free encyclopedia
Jump to navigation Jump to search

Inclinedthrow.gif(400 × 288 pixels, file size: 374 KB, MIME type: image/gif, looped, 102 frames, 10 s)


English: Trajectories of three objects thrown at the same angle (70°). The black object doesn't experience any form of drag and moves along a parabola. The blue object experiences Stokes' drag, and the green object Newton drag.
Date 15 December 2008
Source Own work
Author AllenMcC.
Other versions Inclinedthrow2.gif
GIF developmentCreated with Matplotlib-logo.svg This plot was created with Matplotlib.


# -*- coding: utf8 -*-

import os
import inspect
from math import *
import numpy as np
from scipy.integrate import odeint
from scipy.optimize import newton
import matplotlib as mpl
import matplotlib.pyplot as plt
from matplotlib import animation

# settings
mpl.rcParams['path.snap'] = False
fname = 'inclinedthrow'
size = 400, 288
l, w, b, h = 22.5/size[0], 1-23/size[0], 22.5/size[1], 1-23/size[1]
nframes = 102
delay = 8
lw = 1.
ms = 6
c1, c2, c3 = "#000000", "#0000ff", "#007100"

def projectile_motion(g, mu, pot, xy0, vxy0, tt):
    # use a four-dimensional vector function vec = [x, y, vx, vy]
    def dif(vec, t):
        # time derivative of the whole vector vec
        v = hypot(vec[2], vec[3])
        vxrel, vyrel = vec[2] / v, vec[3] / v
        return [vec[2], vec[3], -mu * v**pot * vxrel, -g - mu * v**pot * vyrel]

    # solve the differential equation numerically
    vec = odeint(dif, [xy0[0], xy0[1], vxy0[0], vxy0[1]], tt)
    return vec[:, 0], vec[:, 1], vec[:, 2], vec[:, 3]  # return x, y, vx, vy

g = 1.
theta  = radians(70)
v0 = sqrt(g/sin(2*theta))
vinf = 2.1
# use identical terminal velocity vinf for both types of friction
mu_stokes = g / vinf**1
mu_newton = g / vinf**2
x0, y0 = 0.0, 0.0
vx0, vy0 = v0 * cos(theta), v0 * sin(theta)

T = newton(lambda t: projectile_motion(g, 0, 0, (x0, y0), (vx0, vy0), [0, t])[1][1], 2*vy0/g)
nsub = 10
tt = np.linspace(0, T * nframes / (nframes - 1), (nframes - 1) * nsub + 1)

traj_free = projectile_motion(g, 0, 0, (x0, y0), (vx0, vy0), tt)
traj_stokes = projectile_motion(g, mu_stokes, 1, (x0, y0), (vx0, vy0), tt)
traj_newton = projectile_motion(g, mu_newton, 2, (x0, y0), (vx0, vy0), tt)

def animate(nframe, saveframes=False):
    print(nframe, '/', nframes)
    t = T * float(nframe) / nframes
    fig.gca().set_position((l, b, w, h))
    plt.xlim(0, 1)
    plt.ylim(0, (h*size[1]) / (w*size[0]))
    plt.xticks([]), plt.yticks([])
    plt.xlabel('Distance', size=12)
    plt.ylabel('Height', size=12)
    plt.plot(traj_free[0][:nframe*nsub+1], traj_free[1][:nframe*nsub+1],
        '-', lw=lw, color=c1)
    plt.plot(traj_free[0][nframe*nsub], traj_free[1][nframe*nsub],
        'ok', color=c1, markersize=ms, markeredgewidth=0)
    plt.plot(traj_stokes[0][:nframe*nsub+1], traj_stokes[1][:nframe*nsub+1],
        '-', lw=lw, color=c2)
    plt.plot(traj_stokes[0][nframe*nsub], traj_stokes[1][nframe*nsub],
        'ok', color=c2, markersize=ms, markeredgewidth=0)
    plt.plot(traj_newton[0][:nframe*nsub+1], traj_newton[1][:nframe*nsub+1],
        '-', lw=lw, color=c3)
    plt.plot(traj_newton[0][nframe*nsub], traj_newton[1][nframe*nsub],
        'ok', color=c3, markersize=ms, markeredgewidth=0)
    if saveframes:
        # export frame
        dig = int(ceil(log10(nframes)))
        fsavename = ('frame{:0' + str(dig) + '}.svg').format(nframe)
        with open(fsavename) as f: content =
        content = content.replace('pt"', 'px"').replace('pt"', 'px"')
        with open(fsavename, 'w') as f: f.write(content)

fig = plt.figure(figsize=(size[0]/72., size[1]/72.))

for i in range(nframes):
    animate(i, True)
os.system('convert -loop 0 -delay ' + str(delay) + ' frame*.svg +dither ' + fname + '.gif')
# keep last frame for two seconds
os.system('gifsicle -k32 --color-method blend-diversity -b ' + fname + '.gif -d' + str(delay) + ' "#0-' + str(nframes-2) + '" -d200 "#' + str(nframes-1) + '"')
for i in os.listdir('.'):
    if i.startswith('frame') and i.endswith('.svg'):


I, the copyright holder of this work, hereby publish it under the following license:
w:en:Creative Commons

attribution share alike

This file is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license.
You are free:
  • to share – to copy, distribute and transmit the work
  • to remix – to adapt the work
Under the following conditions:
  • attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
  • share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.


Add a one-line explanation of what this file represents

Items portrayed in this file


15 December 2008

File history

Click on a date/time to view the file as it appeared at that time.

current16:10, 21 October 2020Thumbnail for version as of 16:10, 21 October 2020400 × 288 (374 KB)Geek3adjusted friction coefficients such to make terminal velocity of both trajectories equal. In this case, the Newton projectile moves further.
12:57, 21 October 2009Thumbnail for version as of 12:57, 21 October 2009400 × 288 (453 KB)AllenMcC.added Newton drag
00:40, 22 December 2008Thumbnail for version as of 00:40, 22 December 2008400 × 299 (393 KB)AllenMcC.== Summary == {{Information |Description={{en|1=Trajectories of two objects thrown at the same angle. The blue object doesn't experience any drag and moves along a parabola. The black object experiences Stokes' drag.}} |Source=Own work by uploader |Author
20:12, 18 December 2008Thumbnail for version as of 20:12, 18 December 2008400 × 299 (393 KB)AllenMcC.== Summary == {{Information |Description={{en|1=Trajectories of two objects thrown at the same angle. The blue object doesn't experience any drag and moves along a parabola. The black object experiences Stokes' drag.}} |Source=Own work by uploader |Author
04:07, 15 December 2008Thumbnail for version as of 04:07, 15 December 2008700 × 519 (636 KB)AllenMcC.{{Information |Description={{en|1=Trajectories of two objects thrown at the same angle. The blue object doesn't experience friction and moves along a parabola. The black object experiences Stokes friction.}} |Source=Own work by uploader |Author=[[User:All

Global file usage

The following other wikis use this file:

View more global usage of this file.