Linear Momentum

Linear momentum, $\vec{p} = m\vec{v}$, is a vector that points the same way as the velocity. It's the natural tool for collisions, where objects push on each other much harder than any outside force does, and explosions, where forces inside a system push its parts apart. Because you only compare the moments just before and just after, you can treat each object as a single point.

The net external force on a system equals its rate of change of momentum, $\vec{F}_{net} = \frac{d\vec{p}}{dt}$, so it is the slope of a momentum–time graph. Impulse, $\vec{J} = \int\vec{F}\,dt$, is the area under a force–time graph, and the impulse–momentum theorem says it equals the change in momentum, $\vec{J} = \Delta\vec{p}$. The same idea covers a system moving at constant velocity while its mass changes, where $\vec{F}_{net} = \frac{dm}{dt}\vec{v}$.

A system's total momentum is the sum of its parts' momenta, and its center of mass moves at $\vec{v}_{cm} = \frac{\sum m_i\vec{v}_i}{\sum m_i}$. If the net external force on the system is zero, total momentum stays constant: by Newton's third law, any momentum one object gains, another object in the system loses. Choosing a system where this holds lets you find velocities just before and after collisions or explosions in one or two dimensions.

In an elastic collision the system's total kinetic energy is the same before and after, even though individual objects can gain or lose kinetic energy. In an inelastic collision some kinetic energy is turned into other forms, like thermal energy and sound, and in a perfectly inelastic collision the objects stick together and move with one velocity. Momentum is conserved in every type as long as no net external force acts.