DOING PHYSICS WITH PYTHON

PYTHON TUTORIAL X-Y PLOTS:

PROJECTILE MOTION

Ian Cooper

matlabvisualphysics@gmail.com

DOWNLOAD DIRECTORIES FOR PYTHON CODE

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Graphs22.py

This article provides information about using Python for plotting X-Y graphs using the example of projectile motion.

PROJECTILE MOTION

It is assumed that the acceleration in the horizontal direction is zero and a constant in the vertical direction. To the right is the +X direction and up is the +Y direction

where g is the acceleration due to gravity. The initial displacement for the projectile launch is given by  x₀ and y₀. The initial launch velocity has a magnitude v₀ and direction φ and the velocity components are

The equations describing the motion are

The code Graphs22.py simulates the motion of two projectiles labelled 1 and 2.

PYTHON

A Figure Window can be used to display the numerical values for the simulation input and output variables. This is often more convenient than displaying the values in the Console Window.

The motion of a projectile can be viewed graphically as shown in the sequence of graphs.

(1)    Multiple plots

(2)   Twin axes

(3)     Multiple subplots

(4)      Multiple subplots of different sizes

(5)    The motion of the projectile can be shown as an animated gif

PYTHON CODE    Graphs22.py

# PROJECTILE MOTION GRAPHICS

import numpy

import numpy as np

import scipy.special

import scipy.special as sps

import math

import matplotlib.pyplot as plt

pi = np.pi

from pylab import figure, plot,show,legend,xlabel,ylabel,title,grid

from pylab import ylim,xlim

from numpy import linspace,sin,cos,exp, zeros, ones

from matplotlib.animation import FuncAnimation, PillowWriter

import time

#%% INPUT PARAMETERS >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>

# time span

tMin = 0; tMax = 8;  Nt = 55

# Initialize arrays: displacements and velocities (magnitude & angle [deg])

x1_0 = 0; y1_0 = 0; v1_0 = 35; phi1_0 = 70

x2_0 = 0; y2_0 = 0; v2_0 = 25; phi2_0 = 40

ax_0 = 0; ay_0 = -9.9

#%%  CALCULATIONS: time, acceleration, velocity, displacements

# Time / velocity / displacement

t = linspace(tMin,tMax, Nt)

Ax = ax_0*ones(Nt)

Ay = ay_0*ones(Nt)

v1x_0 = v1_0*cos(phi1_0*pi/180)

v1y_0 = v1_0*sin(phi1_0*pi/180)

v1x   = v1x_0*ones(Nt)

v2x_0 = v2_0*cos(phi2_0*pi/180)

v2y_0 = v2_0*sin(phi2_0*pi/180)

v2x   = v2x_0*ones(Nt)

v1y = v1y_0 + ay_0*t

v2y = v2y_0 + ay_0*t

x1 = x1_0 + v1x_0*t

x2 = x2_0 + v2x_0*t

y1 = y1_0 + v1y_0*t + 0.5*ay_0*t**2

y2 = y2_0 + v2y_0*t + 0.5*ay_0*t**2

# Max height

ind = np.where(v1y <0)[0][0]-1; tHmax1 = t[ind]

y1Max = max(y1)

ind = np.where(v2y <0)[0][0]-1; tHmax2 = t[ind]

y2Max = max(y2)

# Range

tR1 = 2*tHmax1

xR1 = tR1*v1x_0

tR2 = 2*tHmax2

xR2 = tR2*v2x_0

#%% GRAPHICS

# font1 = {'family':'Tahoma','color':'black','size':12}

# plt.rcParams['font.family'] = ['Tahoma']

plt.rcParams['font.size'] = 12

plt.rcParams["figure.figsize"] = (4,4)

# Fig 1:TRAJECTORY x vs y

fig, ax = plt.subplots(1)

fig.subplots_adjust(top = 0.92, bottom = 0.23, left = 0.20,\

                    right = 0.92, hspace = 0.20,wspace=0.2)

ax.axis('equal')

ax.xaxis.grid()

ax.yaxis.grid()

ax.set_ylabel('y  [m]',color= 'black')

ax.set_xlabel('x  [m]',color = 'black')

ax.set_xlim([0, 101])

ax.set_ylim([-20, 80])

ax.set_xticks(np.arange(0,101,20))

ax.set_yticks(np.arange(-20,81,20))

ax.set_title('Trajectory', fontsize = 12)

ax.text(1, 70, 'y1$_{max}$ = %2.2f m' % max(y1), fontsize = 12, color = 'blue')

ax.text(1, -10, 'y2$_{max}$ = %2.2f m' % max(y2), fontsize = 12, color = 'red')

#fig.tight_layout()

ax.plot(x1,y1,'b',lw = 2, label = '1')

ax.plot(x2,y2,'r',lw = 2, label = '2')

ax.legend()

fig.savefig('a001.png')

#%% Fig 2:  t vs y   t vs y

plt.rcParams["figure.figsize"] = (5,4)

fig, ax = plt.subplots(1)

fig.subplots_adjust(top = 0.94, bottom = 0.23, left = 0.20,\

                    right = 0.82, hspace = 0.20,wspace=0.2)

ax.xaxis.grid()

ax.yaxis.grid()

ax.set_ylabel('y  [m]',color= 'blue')

ax.set_xlabel('t  [s]',color = 'black')

ax.set_xlim([0, 10])

ax.set_ylim([-70, 60])

ax.set_xticks(np.arange(0,11,2))

ax.set_yticks(np.arange(-70,61,20))

ax.tick_params(axis='y', labelcolor="blue")

ax.set_title('v$_0$ = %2.1f m/s  ' % v1_0 + \

             '$  \phi_0$ = %2.1f deg'  % phi1_0,fontsize = 10, color = 'black' )

ax.text(0.2, -60, 'v1$_y$ = 0' + '   y1$_{max}$ = %2.2f m' % max(y1)  \

         , fontsize = 10, color = 'blue')

ax.plot(t,y1,'b',lw = 2)

ax2 = ax.twinx()

ax2.set_ylabel('vy  [m/s]',color= 'red')

ax2.set_xlabel('t  [s]',color = 'red')

ax2.set_xlim([0, 11])

ax2.set_ylim([-70, 60])

ax2.set_xticks(np.arange(0,11,2))

ax2.set_yticks(np.arange(-80,61,20))

ax2.tick_params(axis='y', labelcolor = 'red')

#ax2.yaxis.grid(color = 'r')

ax2.plot(t,v1y,'r',lw = 2)

ax2.plot([0,100],[0,0],'r', lw = 0.5)

fig.legend([' y ',' vy'],loc = 'lower left',ncols = 2 )

fig.savefig('a002.png')

#%%  Fig 3:  4 subplots

plt.rcParams["figure.figsize"] = (7,7)

fig1, axes = plt.subplots(nrows=2, ncols=2)

fig1.subplots_adjust(top = 0.94, bottom = 0.12, left = 0.120,\

                    right = 0.95, hspace = 0.36,wspace=0.40)

axes[0,0].set_title('Trajectorty', fontsize = 10)

axes[0,1].set_title('Displacement', fontsize = 10)

axes[1,0].set_title('Velocity', fontsize = 10)

axes[1,1].set_title('Acceleration', fontsize = 10)

R = 0; C = 0   # x vs y

axes[R,C].set_ylabel('y  [m]',color= 'black',fontsize = 12)

axes[R,C].set_xlabel('x  [m]',color = 'black',fontsize = 12)

axes[R,C].set_xlim([0, 101])

axes[R,C].set_ylim([-20, 80])

axes[R,C].set_xticks(np.arange(0,101,20))

axes[R,C].set_yticks(np.arange(-20,81,20))

axes[R,C].xaxis.grid()

axes[R,C].yaxis.grid()

axes[R,C].text(5, 70, 'v$_0$ = %2.1f m/s  ' % v1_0 + \

             '$  \phi_0$ = %2.1f deg'  % phi1_0,fontsize = 10, color = 'black' )

axes[R,C].plot(x1, y1, 'blue')

R = 0; C = 1   # t vs x,y

axes[R,C].set_ylabel('x & y  [m]',color= 'black',fontsize = 12)

axes[R,C].set_xlabel('t  [s]',color = 'black',fontsize = 12)

axes[R,C].set_xlim([0, 10])

axes[R,C].set_xticks(np.arange(0,11,2))

#axes[0,1].set_ylim([-20, 60])

#axes[0,1].set_yticks(np.arange(-20,81,20))

axes[R,C].xaxis.grid()

axes[R,C].yaxis.grid()

axes[R,C].plot(t, y1, 'blue')

axes[R,C].plot(t, x1, 'red')

R = 1; C = 0  # t vs vx, vy

axes[R,C].set_ylabel('v  [m/s]',color= 'black',fontsize = 12)

axes[R,C].set_xlabel('t  [s]',color = 'black',fontsize = 12)

axes[R,C].set_xlim([0, 10])

axes[R,C].set_xticks(np.arange(0,11,2))

axes[R,C].xaxis.grid()

axes[R,C].yaxis.grid()

axes[R,C].plot(t, v1y, 'blue')

axes[R,C].plot(t, v1x, 'red')

R = 1; C = 1  # t vs ax, ay

axes[R,C].set_ylabel('v  [m/s]',color= 'black',fontsize = 12)

axes[R,C].set_xlabel('t  [s]',color = 'black',fontsize = 12)

axes[R,C].set_xlim([0, 10])

axes[R,C].set_xticks(np.arange(0,11,2))

axes[R,C].xaxis.grid()

axes[R,C].yaxis.grid()

axes[R,C].plot(t, Ay, 'blue')

axes[R,C].plot(t, Ax, 'red')

fig1.savefig('a003.png')

#%%  Fig 4: subplots

plt.rcParams["figure.figsize"] = (6.4,4)

fig3, axes = plt.subplots(1, 2, gridspec_kw={'width_ratios': [1.8, 1]})

fig3.subplots_adjust(top = 0.90, bottom = 0.17, left = 0.150,\

                    right = 0.95, hspace = 0.36,wspace=0.40)

axes[0].set_ylabel('y  [m]',color= 'black',fontsize = 12)

axes[0].set_xlabel('x  [m]',color = 'black',fontsize = 12)

axes[0].set_xlim([0, 101])

axes[0].set_ylim([-20, 80])

axes[0].set_xticks(np.arange(0,101,20))

axes[0].set_yticks(np.arange(-20,81,20))

axes[0].xaxis.grid()

axes[0].yaxis.grid()

axes[0].set_title('x$_R$ = %2.1f m' % xR1 + '   y$_{max}$ = %2.1f m' % max(y1)  \

         , fontsize = 10, color = 'red')

# axes[0].text(25, 70, 'v$_0$ = %2.2f m/s' % v1x_0 + \

#                  '$\phi_0$ deg' % phi1_0,  fontsize = 12, color = 'red')

axes[0].text(10, 70, 'v$_0$ = %2.1f m/s  ' % v1_0 + \

             '$  \phi_0$ = %2.1f deg'  % phi1_0,fontsize = 10, color = 'red' )

axes[0].plot(x1, y1, color='red')

axes[1].xaxis.grid()

axes[1].yaxis.grid()

axes[1].set_ylabel('y  [m]',color= 'blue')

axes[1].set_xlabel('t  [s]',color = 'black')

axes[1].set_xlim([0, 10])

axes[1].set_ylim([-70, 60])

axes[1].set_xticks(np.arange(0,11,2))

axes[1].set_yticks(np.arange(-70,61,20))

axes[1].tick_params(axis='y', labelcolor="blue")

axes[1].set_title('t$_H$ = %2.1f s' % tHmax1 + ' y$_{max}$ = %2.1f m' % max(y1)  \

         , fontsize = 10, color = 'blue')

axes[1].plot(t,y1,'b',lw = 2)

fig3.savefig('a004.png')

#%%   NUMERICAL SUMMARY

plt.rcParams['font.size'] = 12

plt.rcParams["figure.figsize"] = (6,5)

fig, ax = plt.subplots(1)

ax.set_title('PROJECTILE MOTION', fontsize = 10)

ax.set_xlim([0, 101])

ax.set_ylim([0, 120])

ax.set_xticks([])

ax.set_yticks([])

H = 110; h = 10;

ax.text(2,H,'Acceleration: a$_x$ = %2.1f m.s$^{-2}$' % ax_0 + \

        ' a$_y}$ = %2.1f m.s$^{-2}$' % ay_0  , fontsize = 10)

H = H - h   

ax.text(2,H,'(1) Initial Displacement: x$_0$ = %2.1f m' % x1_0 + \

        '    y$_0}$ = %2.1f m' % y1_0  , fontsize = 10, color = [0,0,1])

H = H - h   

ax.text(2,H,'(1) Initial velcocity: v$_{0}$ = %2.1f m.s$^{-1}$' % v1_0 + \

        ' $\phi_0$ = %2.1f deg' % phi1_0  , fontsize = 10, color = [0,0,1])

H = H - h  

ax.text(20,H,'v$_{x0}$ = %2.1f m.s$^{-1}$' % v1x_0 + \

        '   v$_{y0}$ = %2.1f m.s$^{-1}$' % v1y_0  , fontsize = 10, color = [0,0,1])

H = H - h  

ax.text(2,H,'(1) Max height   v$_y$ = 0 ' + '   t = %2.1f s' % tHmax1  + \

        '   y$_{max}$ = %2.1f m' % y1Max  , fontsize = 10, color = [0,0,1])   

H = H - h  

ax.text(2,H,'(1) Range   y = 0 ' + '   t = %2.1f s' % tR1  + \

        '   x$_{R}$ = %2.1f m' % xR1  , fontsize = 10, color = [0,0,1])      

H = H - h   

ax.text(2,H,'(2) Initial Displacement: x$_0$ = %2.1f m' % x2_0 + \

        '    y$_0}$ = %2.1f m' % y2_0  , fontsize = 10, color = [1,0,0])

H = H - h   

ax.text(2,H,'(2) Initial velcocity: v$_{0}$ = %2.1f m.s$^{-1}$' % v2_0 + \

        ' $\phi_0$ = %2.1f deg' % phi2_0  , fontsize = 10, color = [1,0,0])

H = H - h  

ax.text(20,H,'v$_{x0}$ = %2.1f m.s$^{-1}$' % v2x_0 + \

        '   v$_{y0}$ = %2.1f m.s$^{-1}$' % v2y_0  , fontsize = 10, color = [1,0,0])

H = H - h  

ax.text(2,H,'(2) Max height   v$_y$ = 0 ' + '   t = %2.1f s' % tHmax2  + \

        '   y$_{max}$ = %2.1f m' % y2Max  , fontsize = 10, color = [1,0,0])   

H = H - h  

ax.text(2,H,'(2) Range   y = 0 ' + '   t = %2.1f s' % tR2  + \

        '   x$_{R}$ = %2.1f m' % xR2  , fontsize = 10, color = [1,0,0]) 

# fig.savefig('a005.png')   

#%%  Fig 6:  ANIMATION

plt.rcParams['font.size'] = 12

plt.rcParams["figure.figsize"] = (4,4)

# Fig 1:TRAJECTORY x vs y

fig, ax = plt.subplots(1)

fig.subplots_adjust(top = 0.92, bottom = 0.20, left = 0.20,\

                    right = 0.92, hspace = 0.20,wspace=0.2)

ax.axis('equal')

ax.xaxis.grid()

ax.yaxis.grid()

ax.set_ylabel('y  [m]',color= 'black')

ax.set_xlabel('x  [m]',color = 'black')

ax.set_xlim([0, 101])

ax.set_ylim([-20, 80])

ax.set_xticks(np.arange(0,101,20))

ax.set_yticks(np.arange(-20,81,20))

ax.plot(x1,y1,'b',lw = 2, label = '1')

ax.plot(x2,y2,'r',lw = 2, label = '2')

txt = ax.text(5,70, '')

txtT = ax.set_title('Trajectory', fontsize = 12)

line1, = ax.plot([], [], 'bo', ms = 8)    

line2, = ax.plot([], [], 'ro', ms = 8)

ax.grid('visible')

def init(): 

   line1.set_data([], [])

   line2.set_data([], [])

   txt.set_text('')

   # txtT.set_text('')

   return  line1, line2, txt

def animate(n):

     u1 = x1[n]

     v1 = y1[n]

     line1.set_data([u1], [v1])

     u2 = x2[n]

     v2 = y2[n]

     line2.set_data([u2], [v2])

     txt.set_text('time = %.2f' % t[n] + ' s') 

     #txtT.set_text('time = %.2f' % t[n] + ' s') 

     time.sleep(0.1)

     return   line1, line2, txt,

anim = FuncAnimation(fig, animate, init_func = init,

                     frames = Nt, interval = 20, blit = True, repeat = False)

anim.save('ag_005.gif', fps=10)