Forest UCM EnergyIntPart

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Energy of Interacting particles


Translational invariance

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]

Forest_UCM_Energy#Energy_of_Interacting_Particles