Elastic Collision (1D)

Fu-Kwun Hwang

Administrator
Hero Member

Offline

Posts: 3086

1 Elastic Collision (1D)

Embed this message

on: February 19, 2010, 07:19:56 pm »

This is a discussion of elastic collision in one dimension.
Before collision: two particles with mass and velocity as $m_1,\vec{v_1}$ and $m_2,\vec{v_1}$
After collision: the velocity have been changed to $\vec{v_1}'$ and $\vec{v_2}'$

Assume there is no external force or the interval is very short, then
total linear momentum is conserved: i.e. $m_1\vec{v_1}+m_2\vec{v_2}=m_1\vec{v_1}'+m_2\vec{v_2}'$
so $m_1(\vec{v_1}-\vec{v_1}')+m_2(\vec{v_2}-\vec{v_2}')=0$

For elastic collision, the total energy is also conserved. $\frac{1}{2}m_1 v_1^2+\frac{1}{2}m_2 v_2^2==\frac{1}{2}m_1 v_1^{'2}+\frac{1}{2}m_2 v_2^{'2}$
It can be re-write as $\frac{1}{2}m_1(v_1^2-v_1^{'2})= -\frac{1}{2}m_2(v_2^2-v_2^{'2})$
It is the same as m_1(v_1-v_1')(v_1+v_1')= -m_2 (v_2-v_2')(v_2+v_2')
Since $m_1(v_1-v_1')=-m_2 (v_2-v_2')$ , so $(v_1+v_1')=(v_2+v_2')$ or $v_1-v_2=v_2'-v_1'$
The result is
$v'_1= \frac{m_1-m_2}{m_1+m_2} v_1 +\frac{2m_2}{m_1+m_2}v_2=V_{cm}+\frac{m_2}{m_1+m_2}(v_2-v_1)=2V_{cm}-v_1$
and $v'_2=\frac{2m_1}{m_1+m_2}v_1+\frac{m_2-m_1}{m_2+m_1}v_2=V_{cm}+\frac{m_1}{m_1+m_2}(v_1-v_2)=2V_{cm}-v_2$
where $V_{cm}=\frac{m_1V_1+m_2V_2}{m_1+m_2}$
and $V_{cm}=\frac{v_1+v_1'}{2}=\frac{v_2+v_2'}{2}$ or $v_1+v_1'=v_2+v_2'=2V_{cm}$

It means that from the coordinate of center of mass: $V_{cm}=0$ , it reduced to
$v_1'=-v_1$ and $v_2'=-v_2$

Define $\rho=\frac{m_2}{m_1}$ , the above equations can be re-write as
$\frac{v'_1}{v_1}=\frac{1-\rho}{1+\rho}+ \frac{2\rho}{1+\rho}\frac{v_2}{v_1}=2\frac{V_{cm}}{v_1}-1$
$\frac{v'_2}{v_1}=\frac{2}{1+\rho}-\frac{1-\rho}{1+\rho}\frac{v_2}{v_1}=2\frac{V_{cm}}{v_1}-\frac{v_2}{v_1}$

The following simulation plot the above two functions.
The X-axis is $\frac{v_2}{v_1}$ , it range from Vscale*xmin to 1. (There is no collision if $v_2>v_1$ )
The blue curve is $\frac{v_1^'}{v_1}$ and red curve is $\frac{v_2^'}{v_1}$
You can change the ratio of $\frac{m_2}{m_1}$ with slider.

The default value is $\frac{m_2}{m_1}=1$ , so
$v_2^'= v_1$ , so $\frac{v_2^'}{v_1}=1$ is a horizontal line
$v_1^'=v_2$ , so $v_2$ is a straight line with slope 1 (function of $\frac{v_2}{v_1}$ )

Special case:
if $m_1<<m_2$ , $v_1'=-v_1+2v_2$ and $v_2'=v_2$ ,-*-
if $v_2=0$ , then $v_1'=-v_1$ and $v_2'=0$
e.g. a ball hit the wall, it will biunced back with almost the same speed (but in oppositive direction).
if $m_1>>m_2$ , $v_1'=v_1$ and $v_2' =2 v_1-v_2$ ,
if $v_2=0$ , then $v_1'=v_1$ and $v_2'=2 v_1$ ,
e.g. a speedy car hit you while you stand still, you will be kicked by twice the velocity of the car.

Embed a running copy of this simulation

Embed a running copy link(show simulation in a popuped window)

Full screen applet or Problem viewing java?Add http://www.phy.ntnu.edu.tw/ to exception site list
Press the Alt key and the left mouse button to drag the applet off the browser and onto the desktop. This work is licensed under a Creative Commons Attribution 2.5 Taiwan License

Please feel free to post your ideas about how to use the simulation for better teaching and learning.
Post questions to be asked to help students to think, to explore.
Upload worksheets as attached files to share with more users.

Let's work together. We can help more users understand physics conceptually and enjoy the fun of learning physics!

collision1Dplot.gif (8.53 KB, 525x410 - viewed 721 times.)
	Logged

Fu-Kwun Hwang

Administrator
Hero Member

Offline

Posts: 3086

2 Re: Elastic Collision (1D) with EJS event

Embed this message

Reply #1 on: June 18, 2011, 07:45:23 pm »

The following is 1D collision using EJS event

Embed a running copy of this simulation

Embed a running copy link(show simulation in a popuped window)

Please feel free to post your ideas about how to use the simulation for better teaching and learning.
Post questions to be asked to help students to think, to explore.
Upload worksheets as attached files to share with more users.

Let's work together. We can help more users understand physics conceptually and enjoy the fun of learning physics!


	Logged

Fu-Kwun Hwang

Administrator
Hero Member

Offline

Posts: 3086

3 Re: Elastic Collision (２D) processed with EJS event

Embed this message

Reply #2 on: June 18, 2011, 09:56:15 pm »

Here is 2D collision processed with EJS event.
You can lean how to use EJS event to process collision.
Download the jar file, double click to run it, then
right click mouse and select "open ejs model" to view how it was created with EJS.
(You need to download EJS and installed it -- i.e. unzip downloaded ZIP file.)

Embed a running copy of this simulation

Embed a running copy link(show simulation in a popuped window)

Please feel free to post your ideas about how to use the simulation for better teaching and learning.
Post questions to be asked to help students to think, to explore.
Upload worksheets as attached files to share with more users.

Let's work together. We can help more users understand physics conceptually and enjoy the fun of learning physics!


	Logged

Fu-Kwun Hwang

Administrator
Hero Member

Offline

Posts: 3086

4 Re: Elastic Collision (1D)

Embed this message

Reply #3 on: April 13, 2013, 04:54:31 pm »

You are welcomed to check out Elastic 1D collision inquiry


	Logged

Pages: [1] Go Up

Know yourself. Questioning. ...Socrates (469-399 BC)

« previous next »

Related Topics
	Subject	Started by	Replies	Views	Last post
	Ejs open source java applet 1D collision carts Elastic and Inelastic Collision « 1 2 3 » Collaborative Community of EJS	lookang	78	104449	January 16, 2018, 05:00:07 pm by ufabet
	Elastic collision and in-elastic collision Dynamics	Fu-Kwun Hwang	0	10737	May 24, 2009, 04:10:14 pm by Fu-Kwun Hwang
	1 D collision carts Elastic and Inelastic Collision dynamics	ahmedelshfie	6	16365	April 27, 2010, 02:16:18 am by ahmedelshfie
	Elastic collision and in-elastic collision dynamics	ahmedelshfie	2	7823	April 27, 2010, 11:22:51 pm by ahmedelshfie
	Physics of collision showing forces at event for 1 dimension Questions related to EJS	lookang	1	4573	March 13, 2012, 10:41:18 pm by Fu-Kwun Hwang

	Author	Topic: Elastic Collision (1D) (Read 14831 times)
0 Members and 1 Guest are viewing this topic. Click to toggle author information(expand message area).