Phase Space of Moving Object Under Decelerating Force

In [1]:

    

#Import packages.
import numpy as np
import matplotlib.pylab as plt
%matplotlib notebook


    

In [2]:

    

#Set folder to save images in.
import os
#os.chdir('Folder/Address/In/Here')


    

In [3]:

    

#Rename the function 'figure' from the 'plt' library (a.k.a. the 'matplotlib.pylab' library) to make it more convenient to use.
fig = plt.figure()

#Set up axes.
ax = fig.add_subplot(1, 1, 1)


    

In [4]:

    

#Define the parameters in the problem. 
#For this phase space, if you are referring to the two earlier problems "Position of Moving Object Under Decelerating Force" and "Velocity of Moving Object Under Decelerating Force," note that here the parameters k, b, and v_0 must be the same.

k = 2
b = 3
v_0 = 15

C = v_0**2 / (v_0**2 + b**2)

#Pretend to plot the function as a smooth curve by plotting a thousand points close together.
t_f = 10
t = np.linspace(0, t_f, 10000)
#Above creates 1000 evenly-spaced locations between 0 and some final time, t_f.

#Define the function for position.
#Assume x > 0 (or use a negative sign in front of the expression for x < 0).
x = ( abs(b) / ( b**2 * k ) ) * (np.arccos( np.sqrt(C) * np.exp(-b**2 * k * t) ) - np.arccos( np.sqrt(C) ))

#Define the function for velocity.
#Assume v > 0 (or use a negative sign in front of the expression for v < 0).
v = ( abs(b) * C * np.exp( -b**2 * k * t ) ) / np.sqrt( 1 - C * np.exp(-2 * b**2 * k * t))

#Plot the function. Here we want to plot position, x, on one axis, and velocity, v, on the other. Position and velocity are both functions of time, t, so each point in time will make a point on the curve. This is called a parametric curve.
#A parametric curve also uses the 'ax.plot' function.
ax.plot(x, v, 'b-', label = 'Phase of Object')

#Label the plot.
fig.suptitle('Phase Space of Object with Decelerating Force $F = -mk(v^3 + a^2v)$')
ax.set_xlabel('Position, x (meters)')
ax.set_ylabel('Velocity, v ($\\frac{meters}{s}$)')

plt.show()