section_path: Definition in classical mechanics/Many particles
section_name: Many particles
paragraph_text: Many particles
  
  The momentum of a system of particles is the vector sum of their momenta. If two particles have respective masses m and m, and velocities v and v, the total momentum is p = p 1 + p 2 = m 1 v 1 + m 2 v 2. {\displaystyle {\begin{aligned}p&=p_{1}+p_{2}\\&=m_{1}v_{1}+m_{2}v_{2}\,.\end{aligned}}} The momenta of more than two particles can be added more generally with the following: p = ∑ i m i v i. {\displaystyle p=\sum _{i}m_{i}v_{i}.}
  
  A system of particles has a center of mass, a point determined by the weighted sum of their positions: r cm = m 1 r 1 + m 2 r 2 + ⋯ m 1 + m 2 + ⋯ = ∑ i m i r i ∑ i m i. {\displaystyle r_{\text{cm}}={\frac {m_{1}r_{1}+m_{2}r_{2}+\cdots }{m_{1}+m_{2}+\cdots }}={\frac {\sum _{i}m_{i}r_{i}}{\sum _{i}m_{i}}}.}
  
  If one or more of the particles is moving, the center of mass of the system will generally be moving as well (unless the system is in pure rotation around it). If the total mass of the particles is m {\displaystyle m}, and the center of mass is moving at velocity v, the momentum of the system is:
  
  p = m v cm. {\displaystyle p=mv_{\text{cm}}.}
  
  This is known as Euler's first law.