Kinematics from Numerical Integration

Launch a particle from ground level with some speed and some angle and compute the trajectory using numerical integration. This is a very simple example for those learning to program.

import math

# Constants

g = 10.0          # gravity
v0 = 20.0         # initial speed
theta0 = math.pi/4.0   # initial angle

dt = 10.0**(-5.0)   # time step

#######################################

# Initialize simulation

t = 0.0     # time
count = 0   # count for printing

x = 0.0     # initial position
y = 0.0

xd = v0*math.cos(theta0)  # initial velocity
yd = v0*math.sin(theta0)

xdd = 0.0    # initial acceleration
ydd = -g

#######################################

# Run simulation

while y >= 0.0:   # run while object in flight

                    # numerical integration
                    # explicit Euler

    x = x + xd*dt   # update position based on velocity
    y = y + yd*dt

    xd = xd + xdd*dt  # update velocity based on acceleration
    yd = yd + ydd*dt

    xdd = 0.0         # constant accleration
    ydd = -g

    t = t + dt        # advance time and count
    count = count + 1

    if count % 1000 == 0:   # print once every thousand simulation intervals
        print t,x,y

The results were plotted in Excel

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Remember to wrap math in $ ... $ or \[ ... \] to ensure proper formatting.

2 \times 3

2 \times 3

2^{34}

2^{34}

a_{i-1}

a_{i-1}

\frac{2}{3}

\frac{2}{3}

\sqrt{2}

\sqrt{2}

\sum_{i=1}^3

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\sin \theta

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\boxed{123}

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print "import math #constant g = 10.0 #gravity v0 = 20.0 #initial speed theta0 = math.pi/4.0 #initial speed dt = 10.0**(-5.0) #time step ########################################## #Initial simulation t = 0.0 #time count = 0 #count for printing x = 0.0 #initial position y = 0.0 xd = v0*math.cos(theta0) #initial velocity yd = v0*math.sin(theta0) xdd = 0.0 #initial acceleration ydd = -g ############################################## #Run simulation while y >= 0.0: #run while object in flight #numerical integration #elpicit euler x = x + xd*dt #update position based on velocity y = y + yd*dt xd = xd + xdd*dt #update velocity based on acceleration yd = yd + ydd*dt xdd = 0.0 #constanrt acceleration t = t + dt #advance time and count count = count + 1 if count % 1000 == 0: #print once every thousand simulation intervals print (t,x,y)"

Easy Math Editor

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Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

@Steven Chase thanks sir

A Former Brilliant Member - 11 months, 4 weeks ago

@Steven Chase sir i have also typed the the whole code ,but my code is not expressed as your code is expressed in brilliant ??

You need three ticks and then "Python", and then code, followed by three more ticks

Steven Chase - 11 months, 4 weeks ago

@Steven Chase sir can you please show a photo of your excel ,i am facing a bit of difficulty while plotting it in excel.

You have to import the data from a text file and choose space-delimited format

@Steven Chase i stucked here

@A Former Brilliant Member – Have have to import the data in space delimited format, which you haven't done. Then highlight columns B and C.

Insert - Scatter - Scatter with smooth lines

@Steven Chase – @Steven Chase which option should I choose here

@A Former Brilliant Member – space delimited

@Steven Chase – @Steven Chase which option

@A Former Brilliant Member – Just hit finish

@Steven Chase – @Steven Chase now where I have to go?

@A Former Brilliant Member – Highlight columns B and C and then go to:

Insert - scatter chart - scatter with smooth lines

I was going to that scatter only by mistake my arrow is at (other charts)

@Steven Chase

There it is

@Steven Chase yes, Thank you sir
I don't think that anyone in the universe will help me like this you have helped me.

@A Former Brilliant Member – You're welcome. And the cool part is that nowhere in the code is any parabola explicitly specified or coded. It just comes out of the numerical integration

@Steven Chase – @Steven Chase yeah it is interesting

@Steven Chase sir in the last 2nd step what is the meaning of that double equal to = = ??

I single equals sign is used to declare the value of a quantity. A double equals sign is used to compare two quantities to see if they are equal

@Steven Chase okay! thanks , can you please give me some questions or post for practicing numerical integration .
if you post it should be of E and M.
THANKS IN ADVANCE

Wow you're such a dedicated student. Really awesome to see passion and dedication :)

Krishna Karthik - 11 months, 3 weeks ago

@Steven Chase I want to learn monte Carlo, how can I learn??

Hey! You can check out this book.

Aaghaz Mahajan - 11 months, 4 weeks ago

@Aaghaz Mahajan Thanks bro for this book

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$