Forest UCM NLM GalileanTans - Revision history

Foretony: /* Scattering angles */

2014-08-20T13:38:36Z

Scattering angles

Foretony: /* Scattering angles */

2014-08-20T13:34:37Z

Scattering angles

Foretony: /* Find the velocity of the center of momentum (mass) frame */

2014-08-20T13:34:00Z

Find the velocity of the center of momentum (mass) frame

Foretony: /* Find the velocity of the center of momentum (mass) frame */

2014-08-20T13:26:42Z

Find the velocity of the center of momentum (mass) frame

Foretony: /* Find the velocity of the center of momentum (mass) frame */

2014-08-20T13:25:07Z

Find the velocity of the center of momentum (mass) frame

Foretony: /* Find the velocity of the center of momentum (mass) frame */

2014-08-20T13:23:15Z

Find the velocity of the center of momentum (mass) frame

Foretony: /* Galiean Transformation to CM frame */

2014-08-20T13:17:12Z

Galiean Transformation to CM frame

Foretony: /* Galiean Transformation to CM frame */

2014-08-20T13:09:25Z

Galiean Transformation to CM frame

Foretony: /* Galiean Transformation to CM frame */

2014-08-20T13:08:14Z

Galiean Transformation to CM frame

Foretony: /* Galiean Transformation to CM frame */

2014-08-20T13:05:21Z

Galiean Transformation to CM frame

← Older revision		Revision as of 13:38, 20 August 2014
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	: <math>\tan(\psi) = \frac{\sin(\theta)}{\cos(\theta) + \frac{m_1}{m_2}}</math>		: <math>\tan(\psi) = \frac{\sin(\theta)}{\cos(\theta) + \frac{m_1}{m_2}}</math>

		+
		+	Now you can transform the angles between the two frames if you know the masses of the two particles.

	[[Forest_UCM_NLM#Galilean_Transformations]]		[[Forest_UCM_NLM#Galilean_Transformations]]

← Older revision		Revision as of 13:34, 20 August 2014
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	= \frac{\sin(\theta)}{\cos(\theta) + \frac{v_{CM}}{v_1^{\prime}}}</math>		= \frac{\sin(\theta)}{\cos(\theta) + \frac{v_{CM}}{v_1^{\prime}}}</math>

		+
		+
		+	For an elastic collision only the directions change in the CM Frame: <math>u_1^{\prime}= v_1^{\prime}</math> & <math>u_1^{\prime}= v_2^{\prime}</math>
		+
		+	;From the definition of the C.M.
		+	;<math>\vec{v}_{CM} = \frac{m_1 \vec{u}_1 + m_2 \vec{u}_2}{m_1+m_2} = \frac{m_1}{m_1+m_2} \vec{u}_1</math>
		+
		+	;conservation of momentum in CM Frame <math>\Rightarrow</math> :
		+	:<math>m_1 u_1^{\prime} = - m_2 u_2{\prime}</math>
		+
		+	:<math> \Rightarrow v_1^{\prime} = u_1^{\prime} = \frac{-m_2}{m_1} u_2^{\prime}</math>
		+
		+	; Gallilean Coordinate transformation:
		+	;<math>\vec{u}_1 = \vec{u}_1^{\prime} + \vec{v}_{CM} = \vec{u}_1^{\prime} + \frac{m_1}{m_1+m_2} \vec{u}_1</math>
		+	:<math>\Rightarrow u_1{\prime} = \left [ 1 - \frac{m_1}{m_1 + m_2} \right ] u_1 = \frac{m_2}{m_1+m_2}u_1</math>
		+	:<math>\Rightarrow v_1^{\prime} = u_1^{\prime} =\frac{m_2}{m_1+m_2} u_1</math>
		+
		+	; another expression for <math>\psi</math>
		+
		+	using the above gallilean transformation we can do the following
		+
		+	:<math>\frac{v_{CM}}{v_1^{\prime}}= \frac{\frac{m_1}{m_1+m_2} u_1}{\frac{m_2}{m_1+m_2} u_1} = \frac{m_1}{m_2}</math>
		+
		+	or
		+
		+	: <math>\tan(\psi) = \frac{\sin(\theta)}{\cos(\theta) + \frac{m_1}{m_2}}</math>


	[[Forest_UCM_NLM#Galilean_Transformations]]		[[Forest_UCM_NLM#Galilean_Transformations]]

← Older revision		Revision as of 13:34, 20 August 2014
Line 86:		Line 86:

	;The velocity of the center of momentum frame may be calculated (non-relativistically) as the velocity of the center of mass.		;The velocity of the center of momentum frame may be calculated (non-relativistically) as the velocity of the center of mass.
		+
		+	==Scattering angles==
		+
		+
		+	Transforming the scattering angle between reference frames is calculated using the Galilean transformation described by the vector addition of vectors <math>v_1</math> and <math>v_1^\{prime}</math>
		+
		+
		+	[[File:SPIM_ElasCollis_Lab_CM_Frame_Velocities.jpg]]
		+
		+
		+	<math>v_1 \sin(\psi) = v_1^{\prime} \sin(\theta)</math>
		+
		+
		+	<math>v_1 cos(\psi) = v_{cm} + v_1^{\prime} \cos(\theta)</math>
		+
		+	solving for <math>\psi</math>
		+
		+	<math>\tan(\psi) = \frac{\sin(\psi)}{\cos(\psi)} = \frac{v_1^{\prime} \sin(\theta)/v_1}{\frac{v_{CM}}{v_1} + \frac{v_1^{\prime} \cos(\theta)}{v_1} }
		+	= \frac{\sin(\theta)}{\cos(\theta) + \frac{v_{CM}}{v_1^{\prime}}}</math>
		+


	[[Forest_UCM_NLM#Galilean_Transformations]]		[[Forest_UCM_NLM#Galilean_Transformations]]

← Older revision		Revision as of 13:26, 20 August 2014
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	: <math>\vec{v}_{cm} = \frac{m_1 \vec{u}_1 + m_2 \vec{u}_2}{\left ( m_1 + m_2 \right ) }</math>		: <math>\vec{v}_{cm} = \frac{m_1 \vec{u}_1 + m_2 \vec{u}_2}{\left ( m_1 + m_2 \right ) }</math>
		+
		+
		+	The above equation is just the calculation of the velocity of the center of mass of the two particle system.
		+
		+	;The velocity of the center of momentum frame may be calculated (non-relativistically) as the velocity of the center of mass.


	[[Forest_UCM_NLM#Galilean_Transformations]]		[[Forest_UCM_NLM#Galilean_Transformations]]

@@ Line 78: / Line 78: @@
 :: <math>m_1 \vec{u}_1^{\prime} + m_2 \vec{u}_2^{\prime} = 0</math>
 :: <math>m_1 \left ( \vec{u}_1 - \vec{v}_{cm} \right ) + m_2 \left ( \vec{u}_2 - \vec{v}_{cm} \right )= 0</math>
+: <math>\vec{v}_{cm}  = \frac{m_1 \vec{u}_1  + m_2 \vec{u}_2}{\left ( m_1 + m_2 \right ) }</math>
 [[Forest_UCM_NLM#Galilean_Transformations]]

@@ Line 71: / Line 71: @@
 By definition the velocity of the center of momentum is
-\vec{u}_1 = \vec{v}_{cm} + \vec{u}_1^{\prime}
+:<math>\vec{u}_1 = \vec{v}_{cm} + \vec{u}_1^{\prime}</math>
 [[Forest_UCM_NLM#Galilean_Transformations]]

@@ Line 39: / Line 39: @@
 =Galiean Transformation to CM frame=
-Consider the elastic scattering of two hard spheres from a fixed reference frame (Lab Frame) and an inertial frame moving at a speed <math>\vec{v}_{cm}</math> such that the velocities of the two spheres are equal in magnitude but opposite in direction.
+Consider the elastic scattering of two hard spheres from a fixed reference frame (Lab Frame) and an inertial frame moving at a speed <math>\vec{v}_{cm}</math> such that the momentum of the two spheres are equal in magnitude but opposite in direction.
-;A feature of elastic collisions described in the center of mass frame is that the magnitude of each particles momentum remains the same before and after a collision, only their directions change.
+;A feature of elastic collisions described in the center of momentum (mass) frame is that the magnitude of each particles momentum remains the same before and after a collision, only their directions change.
 [[Image:SPIM_ElasCollis_Lab_CM_Frame.jpg | 500 px]]
@@ Line 67: / Line 67: @@
 :<math>\vec{R}</math> = a position vector pointing from the fixed reference frame origin to the moving frame origin
 [[Forest_UCM_NLM#Galilean_Transformations]]

← Older revision		Revision as of 13:09, 20 August 2014
Line 40:		Line 40:

	Consider the elastic scattering of two hard spheres from a fixed reference frame (Lab Frame) and an inertial frame moving at a speed <math>\vec{v}_{cm}</math> such that the velocities of the two spheres are equal in magnitude but opposite in direction.		Consider the elastic scattering of two hard spheres from a fixed reference frame (Lab Frame) and an inertial frame moving at a speed <math>\vec{v}_{cm}</math> such that the velocities of the two spheres are equal in magnitude but opposite in direction.
		+
		+	;A feature of elastic collisions described in the center of mass frame is that the magnitude of each particles momentum remains the same before and after a collision, only their directions change.

	[[Image:SPIM_ElasCollis_Lab_CM_Frame.jpg \| 500 px]]		[[Image:SPIM_ElasCollis_Lab_CM_Frame.jpg \| 500 px]]

@@ Line 59: / Line 59: @@
 ; vector definitions
-:<math>\vec{r}_1</math> = a position vector pointing to the location of <math>m_1</math>
+:<math>\vec{r}_1</math> = a position vector pointing to the location of <math>m_1</math> in the fixed reference frame <math>S</math>
-:<math>\vec{r}_2</math> = a position vector pointing to the location of <math>m_2</math>
+:<math>\vec{r}_2</math> = a position vector pointing to the location of <math>m_2</math> in the fixed reference frame <math>S</math>
-:<math>\vec{R}</math> = a position vector pointing to the center of mass of the two ball system
+:<math>\vec{r}_1^{\prime}</math> = a position vector pointing to the location of <math>m_1</math> in the fixed reference frame <math>S^{\prime}</math>
-:<math>\vec{r} \equiv \vec{r}_1 - \vec{r}_2</math> = the magnitude of this vector is the distance between the two masses
+:<math>\vec{r}_2^{\prime}</math> = a position vector pointing to the location of <math>m_2</math> in the fixed reference frame <math>S^{\prime}</math>
 [[Forest_UCM_NLM#Galilean_Transformations]]

@@ Line 55: / Line 55: @@
 :<math>v_2^{\prime}</math>= final velocity of  <math>m_2</math> in C.M. Frame
 :<math>\theta</math>= scattering angle of <math>m_1</math> in C.M. frame after collision
 [[Forest_UCM_NLM#Galilean_Transformations]]