projectile motion simple simulation using numpy matplotlib python

Question

I am trying to graph a projectile through time at various angles. The angles range from 25 to 60 and each initial angle should have its own line on the graph. The formula for "the total time the projectile is in the air" is the formula for t. I am not sure how this total time comes into play, because I am supposed to graph the projectile at various times with various initial angles. I imagine that I would need x,x1,x2,x3,x4,x5 and the y equivalents in order to graph all six of the various angles. But I am confused on what to do about the time spent.

import numpy as np 
import matplotlib.pylab as plot 

#initialize variables 
#velocity, gravity
v = 30 
g = -9.8 
#increment theta 25 to 60 then find  t, x, y
#define x and y as arrays 
theta = np.arange(25,65,5)   

t = ((2 * v) * np.sin(theta)) / g #the total time projectile remains in the #air
t1 = np.array(t) #why are some negative 



x = ((v * t1) * np.cos(theta))
y = ((v * t1) * np.sin(theta)) - ((0.5 * g) * (t ** 2))

plot.plot(x,y)
plot.show()

np.cos and its cousins np.sin and others expect the angle in radians, so start by multiplying theta with np.pi / 180 or use np.radians(). Also, no need to cast t to a np.array t1 since it is already a numpy array. — Bart
– Bart, Commented Dec 11, 2015 at 23:01

Community · Accepted Answer · 2017-02-09 15:10:19Z

5

First of all g is positive! After fixing that, let's see some equations:

$x=v_{0x}t=v_{0}cos(\theta)t\\y=v_{0y}t-\frac{1}{2}gt^{2}=v_{0}sin(\theta)t-\frac{1}{2}gt^{2}$

You know this already, but lets take a second and discuss something. What do you need to know in order to get the trajectory of a particle?

Initial velocity and angle, right? The question is: find the position of the particle after some time given that initial velocity is v=something and theta=something. Initial is important! That's the time when we start our experiment. So time is continuous parameter! You don't need the time of flight.

One more thing: Angles can't just be written as 60, 45, etc, python needs something else in order to work, so you need to write them in numerical terms, (0,90) = (0,pi/2).

Let's see the code:

import numpy as np
import matplotlib.pylab as plot
import math as m
#initialize variables
#velocity, gravity
v = 30
g = 9.8
#increment theta 25 to 60 then find  t, x, y
#define x and y as arrays

theta = np.arange(m.pi/6, m.pi/3, m.pi/36)

t = np.linspace(0, 5, num=100) # Set time as 'continous' parameter.

for i in theta: # Calculate trajectory for every angle
    x1 = []
    y1 = []
    for k in t:
        x = ((v*k)*np.cos(i)) # get positions at every point in time
        y = ((v*k)*np.sin(i))-((0.5*g)*(k**2))
        x1.append(x)
        y1.append(y)
    p = [i for i, j in enumerate(y1) if j < 0] # Don't fall through the floor                          
    for i in sorted(p, reverse = True):
        del x1[i]
        del y1[i]

    plot.plot(x1, y1) # Plot for every angle

plot.show() # And show on one graphic

edited Feb 9, 2017 at 15:10

CommunityBot

11 silver badge

answered Dec 11, 2015 at 23:12

VlS

5863 silver badges13 bronze badges

Sign up to request clarification or add additional context in comments.

4 Comments

Andras Deak -- Слава Україні Over a year ago

Beat me to it!:) I think the maximal t would be needed if you wanted to plot exactly until the point of falling. And you don't need math: np.pi is perfectly fine. Or the other way around, since your solution is not really using numpy: drop numpy and use math and range for theta:)

VlS Over a year ago

:D Thank you for your suggestions and advice. As you can see, noob force is strong in me xD

Andras Deak -- Слава Україні Over a year ago

Don't worry, 1337ness comes with time and practice;) For what it's worth, I'm also still only learning python/numpy.

VlS Over a year ago

That's right. I manly use it to get results, i'm not interested in coding/programming. My current problems are in network theory, with Networkx :/ , and I have to do it for my research. Which means, google the #@!% out of stackflow and find the solution! :D

Community · Accepted Answer · 2017-05-23 12:09:05Z

4

You are making a number of mistakes.

Firstly, less of a mistake, but matplotlib.pylab is supposedly used to access matplotlib.pyplot and numpy together (for a more matlab-like experience), I think it's more suggested to use matplotlib.pyplot as plt in scripts (see also this Q&A).

Secondly, your angles are in degrees, but math functions by default expect radians. You have to convert your angles to radians before passing them to the trigonometric functions.

Thirdly, your current code sets t1 to have a single time point for every angle. This is not what you need: you need to compute the maximum time t for every angle (which you did in t), then for each angle create a time vector from 0 to t for plotting!

Lastly, you need to use the same plotting time vector in both terms of y, since that's the solution to your mechanics problem:

y(t) = v_{0y}*t - g/2*t^2

This assumes that g is positive, which is again wrong in your code. Unless you set the y axis to point downwards, but the word "projectile" makes me think this is not the case.

So here's what I'd do:

import numpy as np 
import matplotlib.pyplot as plt 

#initialize variables 
#velocity, gravity
v = 30 
g = 9.81  #improved g to standard precision, set it to positive
#increment theta 25 to 60 then find  t, x, y
#define x and y as arrays 
theta = np.arange(25,65,5)[None,:]/180.0*np.pi #convert to radians, watch out for modulo division

plt.figure()

tmax = ((2 * v) * np.sin(theta)) / g
timemat = tmax*np.linspace(0,1,100)[:,None] #create time vectors for each angle

x = ((v * timemat) * np.cos(theta))
y = ((v * timemat) * np.sin(theta)) - ((0.5 * g) * (timemat ** 2))

plt.plot(x,y) #plot each dataset: columns of x and columns of y
plt.ylim([0,35])
plot.show()

I made use of the fact that plt.plot will plot the columns of two matrix inputs versus each other, so no loop over angles is necessary. I also used [None,:] and [:,None] to turn 1d numpy arrays to 2d row and column vectors, respectively. By multiplying a row vector and a column vector, array broadcasting ensures that the resulting matrix behaves the way we want it (i.e. each column of timemat goes from 0 to the corresponding tmax in 100 steps)

Result:

edited May 23, 2017 at 12:09

CommunityBot

11 silver badge

answered Dec 11, 2015 at 23:18

Andras Deak -- Слава Україні

35.4k13 gold badges94 silver badges118 bronze badges

2 Comments

mmb_rach Over a year ago

Thanks! Didn't see this until know, but I have a question. How does it know to print each trajectory in a separate color?

Andras Deak -- Слава Україні Over a year ago

@mmb_rach that's the default behaviour of matplotlib.pyplot: subsequent plots are drawn with seperate colours. The colors are chosen according to the so-called color cycle, which you can set by ax.set_color_cycle for a given axes ax, for instance plt.gca().set_color_cycle for the current axes. This has to be issued before plot commands. The above figure also shows the default color cycle: blue-green-red-cyan-purple-yellow-black (-repeat). I believe this is the same as matlab's old default colororder, changed in R2012b.

Collectives™ on Stack Overflow

projectile motion simple simulation using numpy matplotlib python

2 Answers 2

4 Comments

2 Comments

Your Answer

Linked

Hot Network Questions

Collectives™ on Stack Overflow

2 Answers 2

4 Comments

2 Comments

Your Answer

Sign up or log in

Post as a guest

Linked

Related