File:Mplwp ballistic trajectories velocities.svg

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Summary

DescriptionMplwp ballistic trajectories velocities.svg
English: Plot of a ballistic trajectory with air resistance. The trajectory follows the differential equation r → ¨ ( t ) = − k v ⋅ v → − g ⋅ y → ^ {\displaystyle {\ddot {\vec {r}}}(t)=-k\,v\cdot {\vec {v}}-g\cdot {\hat {\vec {y}}}} with initial conditions r → ( 0 ) = 0 → , r → ˙ ( 0 ) = v 0 ( x → ^ cos ⁡ ( α 0 ) + y → ^ sin ⁡ ( α 0 ) ) {\displaystyle {\vec {r}}(0)={\vec {0}},\;{\dot {\vec {r}}}(0)=v_{0}({\hat {\vec {x}}}\cos(\alpha _{0})+{\hat {\vec {y}}}\sin(\alpha _{0}))} .

The parameters are:

  • g = 1 {\displaystyle g=1} , k = 1 {\displaystyle k=1} , α 0 = 45 ∘ {\displaystyle \alpha _{0}=45^{\circ }}
  • The initial velocity takes the values v 0 = 2 {\displaystyle v_{0}=2} , v 0 = 4 {\displaystyle v_{0}=4} , v 0 = 6 {\displaystyle v_{0}=6} , v 0 = 8 {\displaystyle v_{0}=8} , v 0 = 10 {\displaystyle v_{0}=10}
The differential equation is solved numerically using Scipy odeint.
Date
Source Own work
Author Geek3
SVG development
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 The SVG code is valid.
 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 1.0 
#!/usr/bin/python
# -*- coding: utf8 -*-

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

code_website = 'http://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_velocities.svg'
fig = mplwp.fig_standard(mpl)

xlim = 0,2.6; fig.gca().set_xlim(xlim)
ylim = 0,2.6*355/515.; fig.gca().set_ylim(ylim)
fig.gca().xaxis.set_major_locator(mpl.ticker.MultipleLocator(0.4))
fig.gca().yaxis.set_major_locator(mpl.ticker.MultipleLocator(0.4))

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 = 1.0
k = 1.0
alpha0 = pi/4
for v0 in np.linspace(0, 10, 6)[1:]:
    t1 = brentq(lambda t: ballistic(g,k,[0,0],v0,alpha0,[0,t])[1][1],0.1,5)
    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'$v_0=\,{:.0f}$'.format(v0))

mpl.rc('legend', borderaxespad=1.0)
plt.legend(loc='upper left').get_frame().set_alpha(0.9)
plt.savefig(name)
mplwp.postprocess(name)

Licensing

I, the copyright holder of this work, hereby publish it under the following licenses:
GNU head Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled GNU Free Documentation License.http://www.gnu.org/copyleft/fdl.htmlGFDLGNU Free Documentation Licensetruetrue
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  • to share – to copy, distribute and transmit the work
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inception

28 September 2014

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current19:30, 28 September 2014Thumbnail for version as of 19:30, 28 September 2014600 × 400 (29 KB)(username removed)

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