Forest UCM EnergyIntPart

Energy of Interacting particles

Translational invariance

Consider two particles that interact via a conservative force $\vec{F}$

One potential for Both Particles

Both forces from same potential

just take appropriate derivative

Elastic Collisions

Definition

BOTH Momentum and Energy are conserved in an elastic collision

Example

Consider two object that collide elastically

Conservation of Momentum

$\left ( p_1 + p_2 \right ) _{\mbox{initial}} = \left ( p_1 + p_2 \right ) _{\mbox{final}}$

Conservation of Energy

$\left ( T + U \right ) _{\mbox{initial}} = \left ( T + U \right ) _{\mbox{final}}$

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

$U_{\mbox{initial}} = U_{\mbox{final}} = 0 =$ arbitrary constant

Example

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

Conservation of momentum

$m \vec{v}_1 = m \left (\vec{v}_1^{\;\prime} + \vec{v}_2^{\;\prime} \right )$

Conservation of Energy

$\frac{1}{2} m v_1^2 = \frac{1}{2} m \left (v_1^{\;\prime} \right )^2 + \frac{1}{2} m\left ( v_2^{\;\prime} \right )^2$

Square the conservation of momentum equation

$\vec{v}_1 \cdot \vec{v}_1 = \left (\vec{v}_1^{\;\prime} + \vec{v}_2^{\;\prime} \right ) \cdot \left (\vec{v}_1^{\;\prime} + \vec{v}_2^{\;\prime} \right )$

$v_1^2 = \left (v_1^{\;\prime} \right )^2 + \left ( v_2^{\;\prime} \right )^2 + 2 \vec{v}_1^{\;\prime} \cdot \vec{v}_2^{\;\prime}$

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

$v_1^2 = \left (v_1^{\;\prime} \right )^2 + \left ( v_2^{\;\prime} \right )^2$

and you conclude that

$2 \vec{v}_1^{\;\prime} \cdot \vec{v}_2^{\;\prime} = 0 \;\;\;\; \Rightarrow \vec{v}_1^{\;\prime} \perp \vec{v}_2^{\;\prime}$

Forest_UCM_Energy#Energy_of_Interacting_Particles