Chapter 10: Conservation of Momentum

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If you've ever, uh, wrestled with a physics problem, you know Newton's laws are incredibly powerful.

But sometimes,

well, sometimes you run into things that are just mathematically nightmarish.

Think about trying to calculate the exact motion of, say, ten dollars gas molecules bouncing around in a tiny space.

Or even just predicting the dance of three planets under gravity, the infamous three -body problem.

Analytically, it's often impossible.

You ended resorting to approximations, numerical methods.

So the question is, how do physicists get around this?

How do you understand systems with maybe billions of interactions without getting bogged down?

Well, the answer lies in finding powerful general principles.

And today, we're doing a deep dive into exactly that, focusing on chapter 10 of the Feynman Lectures on Physics, Conservation of Momentum.

Our mission really is to unpack this fundamental principle, see how it lets us sidestep all that insane complexity.

Yeah, absolutely.

Feynman really stresses that these conservation laws aren't just handy calculation tricks.

They're fundamental theorems.

They describe these deep repeating patterns in nature, giving us a solid foundation even when the details are chaotic.

So in this deep dive, we're going to explore why momentum is conserved, and maybe more surprisingly, how this idea actually helps us understand what mass is in a very practical way.

Okay, let's unpack this then.

Yeah.

And it all starts, really, with one of Newton's most fundamental laws.

It really does.

We have to start with Newton's third law, action equals reaction.

That's the bedrock for conservation.

The law basically states that if object A pushes on object B, then object B pushes back on object A with a force that's exactly equal in strength, but precisely opposite in direction.

They always come in these perfectly balanced pairs.

Always balanced.

Always.

Now, the crucial step is linking this to momentum.

Remember, force isn't just push or pull.

Technically, it's the rate of change of momentum.

So if these forces between two particles are equal and opposite.

Ah, then the rate at which one particle's momentum changes must be the exact negative of the rate the other one changes, right?

Exactly.

If one gains momentum at a certain rate, the other loses it at precisely the same rate.

They perfectly counteract each other moment by moment.

Okay, so that's the fascinating part.

If you consider those two particles as your whole system and you add up their momentum changes due to this internal interaction.

It sums to zero.

Always.

The internal forces, the push and pull between them, they cannot change the total momentum of the pair combined.

It remains constant.

Whether they're colliding, exploding apart, orbiting,

doesn't matter.

Doesn't matter for the total momentum.

And this isn't just limited to two particles.

You can scale this up.

Think about a system with billions of particles, those gas molecules, maybe.

The total momentum, which is the vector sum of all the individual momenta m on one v, you're all one plus 10 v of a two, 10 plus three v v three a whole and so on.

That sum stays absolutely constant.

Provided.

There's always a condition, isn't there?

Ah, yes.

Provided the system is truly isolated.

That's the key qualifier.

Right.

So isolated means no outside influences.

Precisely.

The internal forces, all the complicated interactions inside the system, they can redistribute momentum between the particles, but they can never change the grand total for the system.

Only an external force, something pushing on the system from the outside,

like friction from the table the box is on or gravity from earth acting on the whole thing.

Only that can change the total momentum.

That distinction between internal and external forces is really critical then.

It is.

The system is sort of immune to its own internal dynamics when it comes to total momentum.

And you know, this idea that the total momentum is conserved regardless of the internal chaos, that actually leads directly to something quite profound that Feynman brings up,

Galilean relativity.

Yes, it's a direct consequence.

It tells us that the fundamental laws of mechanics work exactly the same, whether you're standing still or moving smoothly at a constant speed.

Like juggling on an airplane.

Right.

Assuming no turbulence.

Exactly.

The physics inside the plane, how the balls move, how they bounce, is identical to the physics on the ground.

You can't do a simple mechanical experiment inside that smooth moving frame to tell if you're moving or not.

The conservation laws hold just the same.

So, okay, this conservation law is clearly fundamental.

It works regardless of your motion.

What else can we do with it?

And this is where I think Feynman gets really clever.

He uses conservation of momentum to define mass itself.

Yeah, it's a beautiful piece of reasoning.

Instead of just saying mass is amount of stuff, he gives an operational definition, something you can actually measure and base the concept on.

Imagine you have two objects, let's call them A and B.

They start at rest, maybe squash together with a spring in between or a tiny explosive charge, then bang, you release them, they fly apart.

Okay, so initially total momentum is zero because nothing's moving.

Right.

And crucially, because the forces are internal, the spring or explosion pushing them apart, the total momentum after they fly apart must also be zero.

Makes sense.

Let's start simple, like Feynman does.

What if A and B are identical, say two identical copper blocks?

Then symmetry tells you everything.

They'll fly apart with exactly the same speed, just in opposite directions.

So mema v -a -e -i is equal in magnitude, but opposite in direction to mema v -b -or.

The total momentum, P -A plus P -B -O, remains zero.

Simple.

Okay, that's intuitive.

But what if they're different?

Say A -A is a small chunk of something and B is a big heavy chunk.

Now it gets interesting.

They explode apart.

You'd expect the lighter object, A, to fly off much faster than the heavier object, B, right?

That's reasonable.

And conservation of momentum demands this.

For the total momentum to still be zero, the momentum of A, mema times v -b's, must exactly cancel the momentum of B, m times v -b.

So v -a -a -extus -m -des -v -b -b.

Wait, but if we don't know the masses hamm and name -b -r -br -o beforehand, how does this help define them?

We can measure the velocities, v -a -b -r after the explosion, though.

That's the key.

Measure the velocities.

Since we know the magnitudes of the momentum must be equal, ma -v -a equals m -ba -v -e -b, we can rearrange this.

The ratio of the masses, ma -a -a -b -b, must be equal to the inverse ratio of the speeds, e -v -b -v -a.

Ah, I see.

So we don't need some standard kilogram weight sitting in Paris.

We can define the ratio of two masses just by letting them interact internally and measuring how fast they recoil relative to each other.

Exactly.

Mass, in this operational sense, is a measure of inertia, how much an object resists changing its velocity when pushed by another object.

It's defined by the interaction itself, based on this fundamental conservation law.

It's quite profound.

It really is.

And Feynman illustrates this with collision experiments, too, which are basically the reverse process.

Let's try to visualize one of those he describes.

Imagine an object with mass and nollar moving along, and it hits an identical object, also mass and nollar, which is just sitting there.

What happens in a perfect elastic collision?

In that perfectly symmetrical case, it's a clean transfer.

The first object, the one that was moving, stops dead in its tracks.

Stops completely.

Stops completely.

And the second object, the one that was stationary, moves off with the exact velocity the first one had initially.

Total momentum before equals total momentum after.

Okay, neat.

But what about unequal masses?

He has that figure, 10 to 4.

Imagine the first object has mass dollars, and it hits a stationary object that's heavier, mass two dollars.

Right.

Now, it's not just a simple swap.

The incoming mass dollars doesn't stop.

In fact, it bounces backward.

Backward.

Yes, it rebounds, but with only one third of its original speed.

And the heavier mass, two mollars, which was stationary, it moves forward, but only with two thirds of the original incoming object's speed.

Wow, okay.

So two mollars goes back at 333, and two mollars goes forward at 2v33.

If two dollars is the initial speed, yes.

And if you calculate the total momentum before and the total momentum after, taking direction into account, so v3 plus 2v3, you'll find they are exactly the same.

Conservation holds.

It dictates those seemingly complicated outcomes.

It's like the ultimate rulebook for interactions.

Now, this talk of collisions naturally brings up energy, doesn't it?

Because momentum is always conserved in these isolated systems, but kinetic energy is a bit more tricky.

It is very much so.

And this distinction leads to two important types of collisions we need to talk about.

First, you have what are called elastic collisions.

Elastic, like a perfect bounce.

Pretty much.

In an ideal elastic collision, not only is momentum conserved, but kinetic energy,

the energy of motion is also conserved.

Think of like billiard balls, ideally, or collisions between individual atoms at low energies.

The total kinetic energy before equals the total kinetic energy after.

Okay, but that sounds idealized.

Most real world collisions aren't like that, are they?

No, most are inelastic collisions.

In these, kinetic energy is not conserved.

Some of it gets transformed into other forms of energy.

Like what?

Heat, sound, deformation of the objects.

Think of two cars crashing, there's a lot of noise, crunching metal, heat generated.

That energy had to come from somewhere.

It came from the kinetic energy they had before the crash.

Exactly.

Or imagine throwing a lump of clay at a wall.

It sticks, right?

It deforms.

Almost all the kinetic energy is lost, converted into heat, and permanently changing the clay's shape.

So energy gets lost, or rather transformed.

But what about momentum in those messy inelastic collisions?

And this is the crucial point Feynman makes.

Even in the most inelastic collision imaginable, where things stick together, make noise, get hot, momentum is still conserved.

Always.

Energy conservation might look messy, but momentum conservation is robust.

That is a powerful distinction.

And we see this principle everywhere, right?

Like rocket propulsion.

Absolutely.

A rocket doesn't push against the air, fundamentally.

It works in space where there is no air.

It works by throwing mass, the hot exhaust gas, backward at high speed.

So the backward momentum of the gas.

Must be balanced by an equal and opposite forward momentum gained by the rocket itself.

The total momentum of the system, rocket plus ejected gas, remains constant.

Usually starting from zero if it launched from rest.

It's pure conservation of momentum in action.

Okay, so this law seems incredibly solid.

But you mentioned earlier, what happens when things get really, really fast?

Like approaching the speed of light.

Does classical PMV still hold up?

Ah, well this is where things get even more interesting, leading into relativity.

The amazing thing is the principle of momentum conservation still holds.

It seems to be a truly universal law.

What the formula PMV needs, tweaking.

Exactly.

It turns out that for the law of conservation of momentum to remain true at relativistic speeds, the classical definition isn't quite right.

We find that mass itself isn't constant.

Mass changes with speed.

Effectively, yes.

In objects in NERFA, its resistance to acceleration increases as its speed gets closer to the speed of light.

The actual formula involves the rest mass total dollars and the Lorentz factor.

One matter dollars, score one V2C2.

So the definition of momentum has to be modified to P pi MV M zero V.

It's this relativistic momentum that is actually conserved in nature, especially in high -speed particle collisions.

So the universe insists on conserving momentum, even if it means redefining what momentum is at high speeds.

That's the profound takeaway Feynman highlights.

The conservation law itself is so fundamental that nature adjusts the definitions of mass and momentum to preserve it under all circumstances.

It's not just a rule, it's apparently a core property of spacetime itself.

And it doesn't stop with mechanics, does it?

You find momentum conservation popping up elsewhere, too.

Absolutely.

Feynman mentions its importance in electromagnetism, where fields themselves can carry momentum, and crucially, it remains a cornerstone in quantum mechanics, long after classical physics breaks down.

It's one of the most reliable, universally applicable principles we have.

Okay, let's try to wrap this up then.

We started by seeing how direct application of Newton's laws can get impossibly complex.

Right, the calculation nightmare.

But we found refuge in this incredibly powerful general principle, conservation of momentum, which stems directly from Newton's third law.

Action -reaction being key.

Then we saw how Feynman uses this very principle, through collision and explosion thought experiments, to give us a practical, operational way to define mass itself based on how objects interact.

Measuring mass by recoil,

essentially.

We distinguish between elastic collisions where kinetic energy is conserved, and inelastic ones where it isn't.

But noted momentum is always conserved.

That robustness is vital.

And finally, we touched on how this principle is so fundamental that it holds even at relativistic speeds, forcing a redefinition of momentum itself and its presence across different areas of physics.

Yeah, the real beauty here, as Feynman shows, is that conservation of momentum gives us predictive power without needing to know every tiny detail of the forces involved.

Whether it's atoms colliding or galaxies interacting, the total momentum before has to equal the total momentum after, period.

It provides this incredible constant thread through the apparent chaos.

So here's a final thought for you, listening out there, to chew on.

We know momentum is conserved in all three spatial directions, X, Y, and Z.

And we know the total momentum of a truly isolated system must remain constant.

If we consider the entire universe as the ultimate isolated system,

what does the law of conservation of momentum imply about the total momentum of everything?

Something to ponder.