Difference between revisions of "Forest UCM EnergyIntPart"

From New IAC Wiki
Jump to navigation Jump to search
Line 3: Line 3:
  
 
=Translational invariance=
 
=Translational invariance=
 +
 +
Consider two particles that interact via a conservative force <math>\vec{F}</math>
  
 
=One potential for Both Particles=
 
=One potential for Both Particles=

Revision as of 13:18, 28 September 2014

Energy of Interacting particles


Translational invariance

Consider two particles that interact via a conservative force [math]\vec{F}[/math]

One potential for Both Particles

Both forces from same potential

just take appropriate derivative

Total work given by one potential

Elastic Collisions

Definition

BOTH Momentum and Energy are conserved in an elastic collision

Example


Consider two object that collide elastically

Conservation of Momentum
[math]\left ( p_1 + p_2 \right ) _{\mbox{initial}} = \left ( p_1 + p_2 \right ) _{\mbox{final}}[/math]
Conservation of Energy
[math]\left ( T + U \right ) _{\mbox{initial}} = \left ( T + U \right ) _{\mbox{final}}[/math]

When the initial and final states are far away fromthe collision point

[math]U_{\mbox{initial}} = U_{\mbox{final}} = 0 =[/math] arbitrary constant


Example

Consider an elastic collision between two equal mass objecs one of which is at rest.

Conservation of momentum
[math] m \vec{v}_1 = m \left (\vec{v}_1^{\;\prime} + \vec{v}_2^{\;\prime} \right )[/math]
Conservation of Energy
[math] \frac{1}{2} m v_1^2 = \frac{1}{2} m \left (v_1^{\;\prime} \right )^2 + \frac{1}{2} m\left ( v_2^{\;\prime} \right )^2[/math]


Square the conservation of momentum equation
[math] \vec{v}_1 \cdot \vec{v}_1 = \left (\vec{v}_1^{\;\prime} + \vec{v}_2^{\;\prime} \right ) \cdot \left (\vec{v}_1^{\;\prime} + \vec{v}_2^{\;\prime} \right )[/math]
[math] v_1^2 = \left (v_1^{\;\prime} \right )^2 + \left ( v_2^{\;\prime} \right )^2 + 2 \vec{v}_1^{\;\prime} \cdot \vec{v}_2^{\;\prime} [/math]

compare the above conservation of momentum equation with the conservation of energy equation

[math] v_1^2 = \left (v_1^{\;\prime} \right )^2 + \left ( v_2^{\;\prime} \right )^2[/math]

and you conclude that


[math]2 \vec{v}_1^{\;\prime} \cdot \vec{v}_2^{\;\prime} = 0 \;\;\;\; \Rightarrow \vec{v}_1^{\;\prime} \perp \vec{v}_2^{\;\prime} [/math]

Forest_UCM_Energy#Energy_of_Interacting_Particles