Momentum Conservation

41 Momentum Conservation

Last chapter, we discovered that even in a Universe with only two particles, Newton’s equations become so complicated we can only solve them by plugging them into a computer. However, not all is lost: there is an interesting symmetry in the problem that will help us, namely Newton’s 3rd law: the push of A on B is equal and opposite to that of B on A. In this chapter we will see how Newton’s 3rd law implies conservation of momentum.

First- what is momentum? Momentum is a measure of the total amount of directional motion in an object or system. But what exactly does that mean?

Exercise 41.1: Momentum

To make progress, let’s think through some specific examples.

In summary, momentum is a vector quantity, which makes sense since it must account for directional information. Moreover, momentum must increase with both mass and speed. Use what you have learned to propose a mathematical definition of momentum: $\vec p = ?$ .

We have defined the momentum of an object via $\vec p = m\vec v$ . If we have a whole bunch of objects, then the total momentum is the sum of the momenta of each of those objects,

$\vec p_{\rm tot} = \vec p_1 + \vec p_2 + ... + \vec p_N = \sum_i \vec p_i.$

We now demonstrate that Newton’s 3rd law implies that if a system is only subject to internal forces, then its total momentum is conserved.

Exercise 41.2: Momentum is Conserved

Consider a system comprised of only two particles, labelled “1” and “2”. The only force felt by particle 1 is due to particle 2, and vice-versa. The total momentum of the system is $\vec p_{\rm tot} = m_1\vec v_1 + m_2\vec v_2$ .

A. Calculate $d\vec p_{\tot}/dt$ by using Newton’s 2nd law. Then, use Newton’s 3rd law to simplify your answer.

That is astonishing: Newton’s laws ensure that the total momentum of the system is conserved, even though we don’t know anything about the forces themselves!

Let’s see how this result gets revised in the presence of external forces.

Exercise 41.3: Momentum is not Conserved

Carry out the same derivation as in problem 39.2, but this time, you should assume that in addition to feeling forces due to each other, objects 1 and 2 also feels an external force. E.g. the total force on object 1 will now look like

$F_{\rm tot,\ on\ 1} = \vec F_{\rm on\ 1\ due\ to\ 2} + F_{\rm ext,1},$

where $\vec F_{\rm ext,1}$ is the total external force acting on object 1.

It is not too hard to show that this same derivation can be generalized to a system with N particles, where N is any arbitrary number. That is, our conclusion is always true. Personally, I like rewriting our final result in terms of differentials,

$d\vec p = \vec F_{\rm ext,tot}dt$ ,

as this makes it clear that external forces produces changes in momentum. This is also the most useful way of writing the equation where you have problems in which the mass of an object is changing (e.g. a rocket!). For most of this class, however, the most useful form of this expression is obtained by integrating, in which case we find

$\vec p_{\rm f} - \vec p_{\rm i} = \int_{\rm i}^{\rm f} \vec F_{\rm ext}(t)dt$ .

Here, I dropped the “tot” subscript because it gets cumbersome to keep writing that over and over. We just need to remember we’re talking about the total external force. The integral of the force is

$\vec J_{\rm ext} \equiv \int_{\rm i}^{\rm f} \vec F_{\rm ext}(t)dt$

is called the impulse. If we take the momentum equation, and move the initial momentum to the right hand side of the equation, we arrive at

$\vec p_{\rm f} = \vec p_{\rm i} + \vec J_{\rm ext}$ .

In other words,

Momentum is conserved unless a system experiences an external impulse.

This is the most useful way of writing this equation for the purposes of problem solving, so this is the version we include in our tablets.

At this point, I’d like to highlight the convention I’m using in my physics tablets. On the left hand side I’m including equations that are truly fundamental laws of physics: the stuff that everything is based on. The equations on the right hand side are useful results (e.g. kinematic equations) or models (e.g. our “law of friction”). These are useful, and you should know them, but they are clearly not fundamental laws of nature.

Let’s take a look at conservation of momentum in action.

Exercise 41.4: Recoil

You load up a cannon with a cannonball. The cannonball has wheels so it can easily roll along the floor. That is, you can pretend the floor is frictionless (we will justify this statement later in this book).

We have just learned why guns have recoil. Check it out.

Now here’s one more interesting question. Suppose the cannon is not allowed to recoil, e.g. by locking it in place against the ground. Evidently, the total momentum of the “cannon+cannonball” system is no longer conserved. Explain why.

In our discussion so far, we have only considered systems with two particles. Fortunately, it’s not too hard to generalize our derivation above to a system with N particles. There are no new physical insights in that derivation, so I won’t go through it. However, I invite you to try it. The end result is that the conservation of momentum for a system holds regardless of the number of particles in the system.

To end this chapter, I want to connect back to Newton’s equation starting from our conservation of momentum equation. Let’s see how this works:

Exercise 41.5: Momentum and Newton’s Law

Our conservation of momentum equation reads $d\vec p_{\rm tot} = \vec F_{\rm ext,tot}dt$ . Consider a system comprised of a single particle of fixed mass $m$ . Note that since a particle can’t push itself, all forces are necessarily external, so we can drop the “ext” subscript.

Use the conservation of momentum equation to find an expression that relates the particle’s acceleration to the external force on the particle.

Evidently, Newton’s equation ( $\vec F_{\rm tot}=m\vec a$ ) is included within our conservation of momentum equation. In fact, as you progress in your study of physics, you will find that the conservation of momentum equation — i.e. $d\vec p = \vec Fdt$ — is in fact more fundamental than Newton’s law.

Key Takeaways

41 Momentum Conservation

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