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# 4: Momentum

• 4.1: Center of Mass
So far we’ve only considered two cases - single particles on which a force is acting (like a mass on a spring), and pairs of particles exerting a force on each other (like gravity). What happens if more particles enter the game? Well, then we have to calculate the total force, by vector addition, and total energy, by regular addition.
• 4.2: Conservation of Momentum
Not only does the center of mass of a system of particles obey Newton’s second law of motion, its total momentum does too. When no external forces act on a system of particles, the total momentum of the system is conserved.
• 4.3: Reference Frames
It is often convenient to analyze your system in a frame that moves with the center of mass, known (unsurprisingly), as the center of mass frame. In this frame, the center of mass velocity is identically zero, and again because of conservation of momentum, all other velocities in this frame must sum to zero.  The ‘real-world’ frame with nonzero center of mass velocity is referred to as the lab frame.
• 4.4: Rocket Science
Although designing a rocket that will follow a desired trajectory (say to Ceres, Pluto, or Planet Nine) with great accuracy is an enormous engineering challenge, the basic principle behind rocket propulsion is remarkably simple. It essentially boils down to conservation of momentum, or, equivalently, the observation that the velocity of center of mass of a system does not change if no external forces are acting on the system.
• 4.5: Collisions
Collisions occur when two (or more) particles hit each other.  During a collision, those particles exert forces on each other,  but in general,  there are no external forces acting on the system consisting of the colliding particles. Consequently, the total momentum of all particles involved in the collision is conserved.
• 4.6: Totally Inelastic Collisions
For the case of two particles colliding totally inelastically, conservation of momentum gives: $$m_{1} v_{1}+m_{2} v_{2}=\left(m_{1}+m_{2}\right) v_{\mathrm{f}}$$. If the masses and initial velocities of the particles are known, calculating the final velocity of the composite particle is thus straightforward.
• 4.7: Totally Elastic Collisions
For a totally elastic collision, we can invoke both conservation of momentum and (by definition of a totally elastic collision) of kinetic energy. We also have an additional variable, as compared to the totally inelastic case, because in this case the objects do not stick together and thus get different end speeds.
• 4.8: Elastic Collisions in the COM Frame
We did the calculation in the lab frame, i.e., from the point of view of a stationary observer. We could of course just as well have done the calculation in the center-of-mass (COM) frame of Section 4.3. Within that frame, as we’ll see below, the relation between the initial and final velocities in an elastic collision is much simpler than in the lab frame.
• 4.E: Momentum (Exercises)