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Momentum conservation (2013)

Page history last edited by Joe Redish 5 years, 5 months ago

Class content > Coherent vs Random Motion > Linear Momentum 

 

Prerequisites

 

A hyper-simplified example

Newton's third law tells us that when two objects interact, they exert forces on each other and those forces are equal and opposite:

 

 

for any type of force. When we combine this with the momentum form of N2, something interesting happens. Let's consider an unrealistically simple case just to see what happens. (It's the way we do physics!) Suppose that we have two objects, A and B, and they interact with nothing else except each other (with an unspecified type of force). Then Newton's second law for the two objects become:

We've put the momenta on the left because that's what we want to pay attention to. Now let's do something a little different. Let's consider the AB pair of objects as a single system by adding these two equations together. (Actually, remember that adding two different objects together and treating them as a system played a role in our deciding that Newton's third law might actually be a reasonable thing to assume and test experimentally.) The result is

 

 

But since the two forces are equal and opposite -- whatever kind of force they are -- by N3, this cancels!  The result is interesting:

 

Now you might say, "So what?"  But when something has its derivative equal to 0 that means that it doesn't depend on the variable you are differentiating with respect to, in this case time. This says that the total momentum of the system doesn't change.  But the individual momenta might change dramatically! If the two objects are billiard balls that are colliding they might take off in totally different directions.  Each momentum changes a lot, but the total doesn't!

 

A more realistic analysis

Let's try to make this a little more general. Suppose we have two objects, A and B, that we are going to consider as a "system".  Each will interact with each other by some (unspecified type of) force and with other objects in the world. We'll refer to those objects that are not A or B as "external" to the system.  Then our N2 equations become

 

 

If we add these two together, taking into account that the forces that A and B exert on each other will cancel, we get the result

 

As a result we can now easily see the conditions for us to get the "total momentum of the system doesn't change" result again.

 

If we have a system of two interacting objects such that the net external force on the objects cancel, then the total momentum of the system doesn't change even though the individual momentum might change.

 

We refer to this as the Theorem of Momentum Conservation.  We can easily see that this can be generalized to any number of objects since for every pair, N3 will say that they will cancel when you add the N2 equations together.  The general result is

 

If we have a system consisting of any number of interacting objects such that the net external force on the objects cancel, then the total momentum of the system doesn't change even though the individual momenta might.

 

It's interesting to realize that whether momentum conservation holds depends on what system we are considering. If we only consider the thrown ball, momentum is not conserved for the ball because there is an external force -- gravity.  But if we included the earth (and were able to measure its momentum to an incredible accuracy) then the momentum of the two would be conserved. If we could consider all the objects in the universe (?) it seems like the total momentum would be conserved. (This is not really legitimate since as we get to sizes comparable to the universe non-Newtonian effects become important.) 

 

Total vs change

There are a number of different ways that this result (momentum conservation).  Let's just consider two objects for whom the net external forces cancel.  We then have the result

We can express this in a variety of ways. We can say that if we look at the total momenta at two different times the result for the total is the same.  Or we could say that the change is 0.  For two items this becomes particularly useful since the total of the two stays the same says that they must change in equal but opposite ways.

 

So there are two ways we can express the fact that the momentum of a pair of objects is conserved:

  • The momenta added together at any (initial) time is equal to the sum of the momenta at a later (final) time.
  • That change in one object's momentum over any time interval is equal and opposite to the change in the other object's momentum.
Be careful! This is a place where if you are sloppy or are doing one-step thinking, you can easily mess up. There are examples we will do in which one of the objects has momentum 0 either initially or finally and for that special case the momentum and the change in the momentum are the same.  But that is not usually the case!  Be very careful to distinguish the momentum from its change and to state your momentum conservation carefully, paying attention to whether you are talking about a momentum or a change in momentum.

 

Follow-on

  • Perfectly inelastic collisions

 

  Workout: Momentum conservation

 

Joe Redish 10/18/11

 

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