Center of Mass Momentum and Collision

Centre of mass

$r_{c m} = \frac{\sum m _{i} r _{i}}{\sum m _{i}} = \frac{1}{M} \int r d m$
Standard bodies
semicircular ring $\frac{2 R}{π};$ semicircular disc $\frac{4 R}{3 π};$ hemisphere (solid) $\frac{3 R}{8};$ cone $\frac{h}{4}$ from base
$M a_{c m} = F_{ext},$ $p = M v_{c m}$

$F_{ext} = 0 \Rightarrow \sum m_{i} v_{i} =$ constant
Recoil
$m_{g} v_{g} = - m_{b} v_{b}$
Variable mass (rocket)
$F = v_{r} \frac{d m}{d t},$ $v = u + v_{r} ln \frac{m _{0}}{m}$

Coefficient of restitution
$e = \frac{velocity of separation}{velocity of approach} = \frac{v _{2} - v _{1}}{u _{1} - u _{2}}$
1-D collision
$v_{1} = \frac{( m _{1} - e m _{2} ) u _{1} + ( 1 + e ) m _{2} u _{2}}{m _{1} + m _{2}},$ $v_{2} = \frac{( m _{2} - e m _{1} ) u _{2} + ( 1 + e ) m _{1} u _{1}}{m _{1} + m _{2}}$
Elastic, equal masses
$e = 1, m_{1} = m_{2} :$ velocities exchange
KE lost in a collision
$Δ K = \frac{m _{1} m _{2}}{2 ( m _{1} + m _{2} )} (u_{1} - u_{2})^{2} (1 - e^{2})$
Bouncing ball
$h_{n} = e^{2 n} h$