File:Mplwp ballistic trajectories golfball.svg

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Original file(SVG file, nominally 600 × 400 pixels, file size: 38 KB)

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Description
English: Plot of a trajectory of a golf ball shot with high velocity at an angle of 45°. The trajectory follows the differential equation with initial conditions , .

The parameters are:

  • , , , , ,
  • Trajectories are drawn for different drag coefficients (no drag), (with dimples) and (without dimples)
The second-order ordinary differential equation system is reduced to a first-order system, using velocity components as independent variables. The system is then solved numerically using Scipy odeint.
Date
Source Own work
Author Geek3
SVG development
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This plot was created with mplwp, the Matplotlib extension for Wikipedia plots.
Source code
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mplwp source code

The plot was generated with mplwp 1.7
#!/usr/bin/python
# -*- coding: utf8 -*-

import matplotlib.pyplot as plt
import matplotlib as mpl
import numpy as np
from math import *

code_website = 'https://commons.wikimedia.org/wiki/User:Geek3/mplwp'
try:
    import mplwp
except ImportError, er:
    print 'ImportError:', er
    print 'You need to download mplwp.py from', code_website
    exit(1)

name = 'mplwp_ballistic_trajectories_golfball.svg'
fig = mplwp.fig_standard(mpl)

xlim = 0,680; fig.gca().set_xlim(xlim)
ylim = 0,180; fig.gca().set_ylim(ylim)

from scipy.integrate import odeint
from scipy.optimize import brentq
def ballistic(g, k, xy0, v0, alpha0, tt):
    # use a four-dimensional vector function vec = [x, y, vx, vy]
    def dif(vec, t):
        v = sqrt(vec[2]**2 + vec[3]**2)
        return [vec[2], vec[3], -k*v*vec[2], -g -k*v*vec[3]]
    
    # solve the differential equation numerically
    vec = odeint(dif, [xy0[0], xy0[1], v0*cos(alpha0), v0*sin(alpha0)], tt)
    return vec[:,0], vec[:,1] # return x(tt) and y(tt)

g = 9.81
m = 0.04593
r = 0.04267 / 2.
v0 = 80.0
rho_air = 1.293
alpha0 = radians(45)
drag = [0.0, 0.1, 0.4]
for c_d in drag:
    F_drag = 0.5 * rho_air * c_d * r**2 * pi
    k = F_drag / m
    t1= brentq(lambda t: ballistic(g,k,[0,0],v0,alpha0,[0,t])[1][1],0.1,20)
    t = np.linspace(0, t1, 5001)
    x, y = ballistic(g, k, [0, 0], v0, alpha0, t)
    while len(y) > 1 and y[-2] <= 0.0: x = x[:-1]; y = y[:-1]
    plt.plot(x, y, label=ur'$c_d=\,{:.1f}$'.format(c_d))

mplwp.move_axes(fig, 18, 0)
plt.xlabel('x [m]'); fig.gca().xaxis.set_label_coords(1.00, -0.02)
plt.ylabel('h [m]'); fig.gca().yaxis.labelpad = 2
mpl.rc('legend', borderaxespad=0.0)
plt.legend(loc='upper right')
plt.savefig(name)
mplwp.postprocess(name)

Licensing

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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 4.0 International 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.

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Date/TimeThumbnailDimensionsUserComment
current20:57, 5 January 2019Thumbnail for version as of 20:57, 5 January 2019600 × 400 (38 KB)Geek3 (talk | contribs)User created page with UploadWizard

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