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Summary

Description

English: Plot of a ballistic trajectory with air resistance for a typical skydive from an airplane. The trajectory follows the differential equation

{\ddot {\vec {r}}}(t)=-\mu \,|v|\cdot {\vec {v}}-g\cdot {\hat {\vec {y}}}

with initial conditions

{\vec {r}}(0)={\vec {0}},\;{\dot {\vec {r}}}(0)=v_{0}{\hat {\vec {x}}}

.

The parameters are:

$g=9.81\,\mathrm {m/s^{2}}$ , $v_{\mathrm {max} }=50\,\mathrm {m/s}$ , $\mu =k/m=g/v_{\mathrm {max} }^{2}$
The horizontal starting velocity takes the values $v_{0}=0\,\mathrm {m/s}$ , $v_{0}=17\,\mathrm {m/s}$ , $v_{0}=33\,\mathrm {m/s}$ , $v_{0}=50\,\mathrm {m/s}$

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

8 September 2020

Source

Own work

Author

Geek3

SVG development

The SVG code is valid.

This plot was created with mplwp, the Matplotlib extension for Wikipedia plots.

Category:Valid SVG created with mplwp code#Mplwp%20skydive%20trajectory.svg

Source code

mplwp source code

The plot was generated with mplwp 1.8

#!/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_skydive_trajectory.svg'
fig = mplwp.fig_standard(mpl)
fig.set_size_inches(480. / fig.dpi, 500. / fig.dpi)
mplwp.move_axes(fig, 24, 6)

xlim = -10,320; fig.gca().set_xlim(xlim)
ylim = -340,20; fig.gca().set_ylim(ylim)
fig.gca().set_aspect('equal')
fig.gca().xaxis.set_major_locator(mpl.ticker.MultipleLocator(100))
fig.gca().yaxis.set_major_locator(mpl.ticker.MultipleLocator(100))

from scipy.integrate import odeint
from scipy.optimize import brentq
def ballistic(g, mu, 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], -mu*v*vec[2], -g -mu*v*vec[3]]
    
    # solve the differenctial 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
vmax = 50.
mu = g / vmax**2
alpha0 = 0.

for iv, v0 in enumerate((0, 17, 33, 50)):
    t1 = brentq(lambda t: ballistic(g,mu,[0,0],v0,alpha0,[0,t])[1][1]-ylim[0],0,20)
    t = np.linspace(0, t1, 5001)
    x, y = ballistic(g, mu, [0, 0], v0, alpha0, t)
    plt.plot(x, y, label='$v_0 = {:.0f}\,m/s$'.format(v0), zorder=-iv)
    col = fig.gca().lines[-1].get_color()
    xmax = ballistic(g, mu, [0, 0], v0, alpha0, [0, 100*vmax/g])[0][-1]
    plt.axvline(xmax, 0, 0.33, linestyle='--', dashes=[10,10],
        color=col, zorder=-5, lw=1.5)
    times = np.arange(t1)
    xt, yt = ballistic(g, mu, [0, 0], v0, alpha0, times)
    plt.plot(xt, yt, '.', color=col, label=None, zorder=-iv-0.5)
    
    if v0 == 50.:
        for ti, xi, yi in zip(times, xt, yt):
            plt.text(xi+8*sin(pi/2*ti/10), yi+8*cos(pi/2*ti/10),
                '{:.0f}s'.format(ti), ha='left', va='center', fontsize=10)

plt.xlabel('x [m]')
plt.ylabel('y [m]')
fig.gca().xaxis.labelpad = 0
fig.gca().yaxis.set_label_coords(-0.08, 0.95)
mpl.rc('legend', borderaxespad=0.4)
plt.legend(loc='upper right')

plt.savefig(name)
mplwp.postprocess(name)

Licensing

I, the copyright holder of this work, hereby publish it under the following license:

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.

Category:CC-BY-SA-4.0#Mplwp%20skydive%20trajectory.svg

Category:Self-published work Category:Photos by User:Geek3 Category:Ballistics Category:Trajectories Category:Physics plots Category:Parachuting

Wikimore

File:Mplwp skydive trajectory.svg

Summary

mplwp source code

Licensing