Notes:Collision

Conservation of momentum

Yields:

• [ilmath]v_2'\eq -\frac{m_1}{m_2}v_1'+\frac{1}{m_2}(m_1v_1+m_2v_2)[/ilmath] (written in the form [ilmath]y\eq m\cdot x+c[/ilmath])

As this is the equation for a line (where [ilmath]x\eq v_1'[/ilmath] and [ilmath]y\eq v_2'[/ilmath]) we can see for huge {{|v_1'}} we get huge [ilmath]v_2'[/ilmath] values (but going the other way)

This is because technically if we have 2 particles just next to each other but not moving and suddenly they fly apart in opposite directions - momentum is conserved.

Think for example of a firework, fired vertically there's no momentum in the horizontal plane, hence the relative symmetry of the explosion.

If the firework is misfired and has a "general left-ward trend" shall we say, after it explodes, considering all the pellets as the system, it still has a "left-ward trend", this trend is the [ilmath]\mathbf{+c} [/ilmath] term in the line.

Let us now introduce energy, who's magnitude is based purely on the magnitude of velocity rather than signed like momentum. This will cap extreme situations we get from the huge (and opposite) [ilmath]v_i'[/ilmath]

Conservation of energy

We will consider:

• [ilmath]E_0:\eq \frac{1}{2}m_1v_1^2+\frac{1}{2}m_2v_2^2[/ilmath] to be energy before
• [ilmath]E_1:\eq \frac{1}{2}m_2(v_1')^2+\frac{1}{2}m_2(v_2')^2+W[/ilmath] to be energy after
  • here we can use [ilmath]W[/ilmath] (for Work) to allow us to be able to describe results where we have more energy afterwards then we did before, put in by the explosion of a firework. Or what energy must be taken out to get an observed result
  • It is tempting to consider only [ilmath]W:\eq -L[/ilmath] where [ilmath]L[/ilmath] is for 'latent' energy, energy which is hidden (and has become heat in collisions which don't conserve mechanical energy) however I use a general [ilmath]W[/ilmath] term to allow us to put work in.
  • If we do consider [ilmath]W[/ilmath] as putting work in we must really define it as [ilmath]W_{\text{in} } -L[/ilmath] where we consider the total work in, and what amount was lost due to the imperfect nature of the collisions