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