Momentum Conservation
What is Conservation?
In English, "conservation" is the act of preventing injury, decay, waste, or loss. In physics when we speak of conservation of momentum, we certainly don't imply that momentum can be injured or wasted. It is more the sense of decay or loss that we mean. The idea is to consider the following question: If I observe a system with some momentum right now, will I find a different value later - after things have moved, collided, etc?
The answer is that if all interactions of the system are being considered, then no. We have to be careful, however, to consider all interactions. By interactions we mean forces. The point of this section is that momentum changes due to the application of a force for a duration of time. Mathematically, the relationship is this:
Assumptions
In this section I want to take the time to show you why momentum is conserved in nature. It turns out that it is quite easy to show if we start by assuming that Newton's laws are valid. We will be using both the 2nd and the 3rd laws during the derivation, and this may concern you if you recall me mentioning when the laws were introduced that neither is valid in certain situations that require relativity.
In spite of this, you may take comfort in hearing that the law of momentum conservation may be shown in other ways that indicate that it is a much stronger statement than either of Newton's laws and in fact holds up in every instance in nature. The form we will be deriving, however, will only be good for speeds below around 10% light speed. In third semester we will look at the more general form of the equation.
Derivation
This discussion is a repeat of that found in the dynamics chapter. Given two objects labeled 1 and 2 that exert a force on one another, as mentioned back then, momentum is conserved for the two-body system so long as we don't have external forces from outside the system acting. The math went like this:
Limitations
Recall that all fundamental interactions are only between two entities as far as we know. In any many-body system you can reduce all the interactions to paired interactions. That means given particles A, B, and C there is an AB, BC, and AC interaction.
Regarding momentum conservation, the important thing to remember is that for any system comprising N>0 particles, momentum will be conserved (constant) in the absence of external forces. If there is some external force due to some external object, we can restore momentum conservation by bringing that object into our system when we do the mathematics. Realize that "bringing it into our system" is an intellectual and not a physical procedure. It simply means that we need to include another term for momentum in our accounting.
![[url=https://pixabay.com/en/accident-collision-crash-152075/]"Car Wreck"[/url] by OpenClipart-Vectors is in the [url=http://creativecommons.org/publicdomain/zero/1.0/]Public Domain, CC0[/url]
In an auto accident, we often assume that momentum is conserved.](https://www.geogebra.org/resource/bpJtcuAy/vNfdGyyt1uNyZ2pf/material-bpJtcuAy.png)
Be Careful with Assumptions!
We often assume in situations like car accidents that momentum is conserved. We just mentioned above that momentum is only conserved in the absence of external forces. The trouble with car accidents involving two cars is that the friction force on the tires is often quite large. That friction is the force on the cars due to the earth - the earth not being a part of our system of two cars that collided. Further, it is not practical to bring earth into our system since we want it to be our assumed fixed coordinate system. And if we did want to include it anyway, what velocity vectors would we assign to the planet before versus after the collision? Hmm. Bottom line is that we should expect that using momentum conservation for car wrecks is approximate at best.
EXAMPLE: Let's try it out nonetheless. Suppose the red car above has a mass of 1200kg and an initial velocity vector of and the orange car has a mass of 1400kg and an initial velocity of If the cars get tangled together during the wreck and slide as one mass, what will the velocity of that mass be right after the collision?
SOLUTION: We assume that momentum is conserved, or Expanding the deltas and separating terms gives us
When is Momentum Conservation Realistic?
Scenarios in which we can do a calculation like the one above and expect good agreement with experiments are ones in which external forces are small or negligible as compared with internal forces of interaction. Billiards shots are a decent example because the collision involves forces much bigger than the small rolling resistance force of the balls on the table. Things colliding in space - like a comet with earth - are also good since the only external forces are very small gravitational forces from distant objects. Particle collisions in accelerator labs throughout the world are among the best candidates. These are done in vacuum and other external forces are minuscule comparing to the very large forces of interaction among the particles.
Very violent collisions like those between a baseball and bat, golf ball and club, and other such scenarios are also very good candidates. In these cases, while gravity is certainly a factor, relative to the collision forces gravity is feeble. So a baseball bat hitting a home run will conserve momentum for the ball and bat system much better than the same bat bunting the same ball gently will.