PX155 - F1 - collisions in 1D

conservation of momentum and reference frames

speeds are measured in "lab frame", ie: relative to some stationary equipment
assume:
- momentum is conserved (no external forces)
- potential energy is irrelevant, ie: no long range forces
conserve momentum: $m_{1} u_{1} + m_{2} u_{2} = m_{1} v_{1} + m_{2} v_{2}$ :

p_{1} + p_{2} = q_{1} + q_{2}

consider $v_{c m} = {\dot{r}}_{c m} = \frac{m_{1} u_{1} + m_{2} u_{2}}{m_{1} + m_{2}} = \frac{p_{1} + p_{2}}{m_{1} + m_{2}}$
- or, $v_{c m} = \frac{q_{1} + q_{2}}{m_{1} + m_{2}} = \frac{m_{1} v_{1} + m_{2} v_{2}}{m_{1} + m_{2}}$
- $v_{c m}$ is conserved if momentum is conserved
consider a second observer is moving in the lab at speed $v_{c m}$ . defines COM frame denoted with a dash
- so,
  $u_{1} = u_{1}^{'} + v_{c m} ⟹ u_{1}^{'} = u_{1} - v_{c m}$
  $u_{2} = u_{2}^{'} + v_{c m} ⟹ u_{2}^{'} = u_{2} - v_{c m}$
  - note: $v_{c m}^{'} = 0 = \frac{p_{1}^{'} + p_{2}^{'}}{m_{1} + m_{2}} = \frac{q_{1}^{'} + q_{2}^{'}}{m_{1} + m_{2}}$
- total momentum $= 0$ in COM frame
- only unknown is magnitude of $q_{1}^{'}$ and $q_{2}^{'}$

all magnitude of momenta is lost , $q_{1}^{'} = q_{2}^{'} = 0 ⟹$ particles stick after collision
in lab frame: $v_{1} = v_{2} = v_{c m}$