Conservation of Momentum

In physics, the systems of interest are often full of dynamics, and in Newtonian mechanics, that is certainly the case. When everything is changing from moment to moment: forces, positions, velocities, et cetera, it may come as a surprise that there exist some quantities that never change. There are however a few special quantities that remain constant even as the components of a system move around and explore the space of possible arrangements. One of these is the total momentum, whose conservation is implied by Newton's laws of motion.

As we showed before, if we consider two particles interacting with one another in free space, A and B, particle A feels the force $F_\textrm{AB}$ from B, while B feels the force $F_\textrm{BA}$ from A. In Newton's third law, we showed that $F_\textrm{AB}=-F_\textrm{BA}$.

From the second law, we know that the change in momentum of particle A per unit time is given by $\Delta p_\textrm{A} / \Delta t = F_\textrm{AB}$, and that $\Delta p_\textrm{B} / \Delta t = F_\textrm{BA}$. Now, the total momentum of the system is given by $p_\textrm{A} + p_\textrm{B}$ and the rate at which the total momentum change is given by

$$\frac{\Delta p_\textrm{A}}{\Delta t} + \frac{\Delta p_\textrm{B}}{\Delta t} = \frac{F_\textrm{AB} + F_\textrm{BA}}{\Delta t} = \frac{0}{\Delta t} = 0$$

If something changes at a rate of zero, it isn't changing at all. Hence, the total momentum of the two particles remains constant!

This calculation can be easily extended to systems of $n$ particles interacting with each other. The change in the total momentum of a system of $n$ particles is given by $\displaystyle \sum_i \frac{\Delta p_i}{\Delta t}$.

Now the change in momentum of each particle $\displaystyle \frac{\Delta p_i}{\Delta t}$ is given by the sum of its interactions with all the other particles $\Delta p_i = \Delta t \sum_j F_{ij}$, so that the change in the total momentum of the system of particles is given by $\Delta t \sum_{ij} F_{ij} = \Delta t \frac{1}{2}\sum_{ij} \left(F_{ij} + F_{ji}\right)$.

Because the forces of interaction between any two particles are balanced $\left(F_{ij} =- F_{ji}\right)$, each term $\left(F_{ij} + F_{ji}\right)$ is equal to zero. Therefore, $$\Delta p = \Delta t \frac{1}{2}\sum_{ij} 0 = 0$$ and the total momentum of the system is constant!

The conservation of momentum is useful whenever we analyze collisions, whether they be a bat and a ball, two cars in a crash, or subatomic particles colliding in a particle accelerator, giving rise to exotic new forms of matter. In each one of these cases the conservation of momentum aids us in our investigation.

Momentum conservation in problems

In the case of two cars of mass $M_1$ and $M_2$ colliding, we have: $$M_1v_1^i + M_2v_2^i = M_1v_1^f + M_2v_2^f$$

If you study particle physics, the annihilation between an electron and positron to form two photons the conservation of momentum becomes: $$p_{e^-}^\mu + p_{e^+}^\mu = p_\gamma^\mu + p+\gamma^\mu$$ where the $\mu$ indicates that these are the four-momentum, the generalization of ordinary momentum in relativity.

The crucial property of the system that leads to this result is the fact that it is closed. In other words, the particles under consideration interact with each other and nothing else. A practical definition of a closed system in classical mechanics is one that we can contain within a surface, across which no forces act, i.e. $F_\textrm{ext} = 0$, and no matter passes. For example, in a space where two particles are the only things in existence, we can draw an imaginary ball around them. No forces act across the ball because there is nothing else in the universe.

In the real world, this is of course never the case. No matter how far one gets away from other matter and energy in the universe, there is always some remnant that spoils the "closed'' nature of any real system. However, such external forces and fields can usually be taken to be very small compared to the magnitude of the interactions being studied, so that for all practical purposes, the system is "closed'', and the conservation of momentum still applies, approximately.

Were the system to be acted upon by a external force, or some kind of field, we'd have expand our system to include the source of the external forces if we wanted to preserve the conservation of momentum.

Finally, we point out that momentum depends on the frame of reference of the observer. If someone sees a bullet of mass $m$ fly by with velocity $\vec{v}$, they'll measure the momentum $p=m\vec{v}$, whereas somebody driving in a car that has velocity $\vec{u}$ in the direction of the bullet will measure the momentum $p^\prime = m \left(\vec{v} - \vec{u}\right)$.