Author Topic: Elastic Collision (1D)  (Read 15209 times)

Fu-Kwun Hwang

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Elastic Collision (1D)
« on: February 19, 2010, 08:19:56 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. $m_1vec{v_1}+m_2vec{v_2}=m_1vec{v_1}'+m_2vec{v_2}'$
 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 $ho=frac{m_2}{m_1}$, the above equations can be re-write as
$frac{v'_1}{v_1}=frac{1-ho}{1+ho}+ frac{2ho}{1+ho}frac{v_2}{v_1}=2frac{V_{cm}}{v_1}-1$
$frac{v'_2}{v_1}=frac{2}{1+ho}-frac{1-ho}{1+ho}frac{v_2}{v_1}=2frac{V_{cm}}{v_1}-frac{v_2}{v_1}$

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< 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.

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
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Fu-Kwun Hwang

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Re: Elastic Collision (1D) with EJS event
« Reply #1 on: June 18, 2011, 08:45:23 pm »
The following is 1D collision using EJS event

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Re: Elastic Collision (?D) processed with EJS event
« Reply #2 on: June 18, 2011, 10:56:15 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.)

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
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  • Post questions to be asked to help students to think, to explore.
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Fu-Kwun Hwang

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Re: Elastic Collision (1D)
« Reply #3 on: April 13, 2013, 05:54:31 pm »
You are welcomed to check out  Elastic 1D collision inquiry