You fire a bullet into a block of wood and watch it swing. But then you try the same thing with a magnet, or on a rough floor, or inside a rocket mid-burn — and suddenly the numbers don't add up. Textbook physics says momentum is conserved. So what gives?
Here's the thing — linear momentum isn't some magic rule that always holds. It's a condition-dependent law. And most people walk out of intro physics thinking it's universal when it really, really isn't.
I've read enough half-explained forum threads on this to know the confusion is real. So let's actually dig into when linear momentum is not conserved, and why that's not a failure of physics — it's a failure of how we often explain it Worth keeping that in mind. That alone is useful..
The official docs gloss over this. That's a mistake.
What Is Linear Momentum (And What "Conserved" Actually Means)
Look, momentum is just mass times velocity. Which means a bowling ball rolling slow and a tennis ball flying fast can have the same momentum if the mass-velocity tradeoff lines up. Simple enough.
When we say momentum is conserved, we mean the total momentum of a system stays the same before and after some event — a collision, an explosion, a push. No gain, no loss. The shorthand you'll hear is "momentum is conserved in a closed system with no external forces Easy to understand, harder to ignore..
But that phrase — closed system with no external forces — is doing a lot of heavy lifting. And it's exactly where people get lost Small thing, real impact..
The System Is the Whole Game
Turns out, whether momentum is conserved depends entirely on what you decide to call "the system.Also, " If you include the Earth in your system, then even a falling apple conserves momentum (Earth moves up, apple moves down, tiny but real). Exclude the Earth, and suddenly momentum looks broken because the ground yanks on the apple.
So when someone asks "when is linear momentum not conserved," the honest answer is: whenever your chosen system is open to outside pushes or pulls.
Internal vs External Forces
An internal force is between objects in your system. That said, those cancel in pairs and don't change total momentum. This leads to an external force comes from outside — friction from a floor, gravity from a planet you didn't include, a hand that shoves. Those are the momentum killers.
Why It Matters / Why People Care
Why does this matter? Because most people skip it and then trust the wrong calculation.
In engineering, if you design a braking system assuming momentum stays constant on a skidding surface, you'll be off. Think about it: in forensic crash reconstruction, investigators know a collision on an icy road vs a dry road changes what "conserved" even means for the cars alone. And in particle physics, momentum non-conservation in a narrowed system is how we detect unseen forces or particles we didn't put in the box.
Real talk — the places where momentum isn't conserved are often the most interesting. Day to day, they tell you something is interacting with the world beyond your frame. Miss that, and your model lies It's one of those things that adds up..
What Goes Wrong When You Assume It Always Holds
I know it sounds simple — but it's easy to miss. It isn't lower in the universe. A student calculates a two-cart collision on a track, forgets the track has friction, and gets confused why the final momentum is lower. It's lower in their tiny, open system. The "lost" momentum went into the Earth via the track.
That kind of mistake isn't just academic. It shows up in robotics, ballistics, even video game physics engines that fake conservation and then look wrong to players who intuitively know something's off Which is the point..
How It Works (Or Rather, When It Breaks)
The meaty part. Let's walk through the actual situations where linear momentum is not conserved for the system you're probably looking at.
External Forces Are Present
It's the big one. If a net external force acts on your system, momentum changes. Still, newton's second law, in momentum form, is F = dp/dt. Non-zero F means p changes. Plain and simple.
Examples:
- A block sliding on a rough table. Friction is external. Momentum drops as heat and floor vibration rise. Think about it: - A ball in free fall if you only count the ball. Gravity is external. Momentum grows downward.
- A car accelerating because the engine pushes through the wheels against the road. In real terms, the road is external. Car gains momentum; road+Eart h takes the opposite, usually ignored.
The System Is Not Closed
Say two magnets snap together on a table. They speed up toward each other — momentum of the two-magnet system increases from zero. Even so, wait, how? The table and Earth supplied force through the magnetic field interaction mediated by supports. You watch only the magnets. If you don't include those, momentum isn't conserved in your slice Most people skip this — try not to..
Same with a person jumping. You (alone) gain upward momentum. That said, momentum of "you" is not conserved. You pushed on Earth.
During Explosions With Outside Constraints
A firecracker blows apart in your hand. So the pieces go sideways, but your hand exerts force, so the system of just the cracker isn't free. And include hand and arm, and it balances. Exclude them, and momentum of the fragment cloud isn't conserved No workaround needed..
In Non-Inertial Reference Frames
Here's a subtle one most guides get wrong. In real terms, momentum conservation as we write it assumes an inertial frame — not accelerating. Still, sit in a car taking a hard turn and watch a ball roll. So in your frame, the ball gains momentum sideways for no local reason. That's fake non-conservation from frame acceleration, not physics breaking. But in practice, if you're calculating from that frame, linear momentum of your system isn't conserved unless you add fictitious forces Nothing fancy..
When Fields Carry Momentum You Ignore
Electromagnetic fields carry momentum. Here's the thing — if you count only the particles, momentum isn't conserved during the interaction. A charged particle pushed by another across a room exchanges momentum with the field. Think about it: the field hid some. This is huge in advanced physics and easy to miss in simple models.
Relativistic and Quantum Caveats
At very high speeds, "linear momentum" means gammamv, not just mv. Conservation still holds in closed systems, but if you use the wrong formula and an open frame, you'll think it's broken. In quantum measurements, a position measurement can collapse a state and the expectation of momentum for a subsystem shifts — but again, the measuring device is the external world.
Common Mistakes / What Most People Get Wrong
Honestly, this is the part most guides get wrong. They say "momentum is always conserved" and then quietly add asterisks.
Mistake 1: Forgetting the Earth
Most Earth-bound examples exclude the planet. Then they act shocked momentum "disappears.In practice, " It didn't. You left the biggest object in the room out of the system.
Mistake 2: Counting Only the Visible Objects
A collision inside a box on a scale. Box shakes, scale reads more. Momentum of the contents alone changed because the box wall pushed. People count the balls, not the box Worth keeping that in mind..
Mistake 3: Assuming Frictionless Means Free
Even frictionless, a constraint like a string or a wall is external. A ball on a string swinging isn't conserving linear momentum — the string pulls. It conserves angular momentum around the pivot, totally different beast.
Mistake 4: Mixing Frames Mid-Calculation
You start in the lab frame, then switch to one cart's frame without transforming. Practically speaking, suddenly totals don't match. That's you, not physics Surprisingly effective..
Mistake 5: Thinking Energy and Momentum Break Together
They don't. Momentum can be non-conserved while energy is weird too, but often friction kills momentum (external) while energy just converts to heat (still "conserved" broadly). Confusing the two leads to double errors.
Practical Tips / What Actually Works
Skip the generic advice. Here's what I'd tell a friend actually trying to solve a problem:
- Define your system in one sentence before calculating. "System = cart A, cart B, and the track." If track is attached to Earth, you just included Earth. Good.
- List external forces explicitly. Gravity, normal, friction, hand, wall. If the list isn't empty, momentum of that system isn't conserved. Period.
- If momentum seems off, widen the system. Add the floor. Add the air. Add the field. Nine times out of
ten, the missing momentum is sitting in something you arbitrarily excluded Small thing, real impact..
- Check your frame before and after. If you computed initial momentum in the lab frame, compute final in the lab frame. Don’t drift into “the moving cart’s frame” without a Lorentz or Galilean transform.
- Use the right momentum type. Linear for translation, angular for rotation about a point, canonical for charged particles in fields. Grab the wrong one and the math will lie to you.
- Separate measurement from dynamics. In quantum or sensor-based setups, a reading change is not a violation; it’s the apparatus becoming part of the interaction.
Conclusion
Momentum is not a law that occasionally fails — it is a statement about what you chose to include. When it appears not to be conserved, the failure is almost always in the model: a hidden external force, an excluded massive body, a silent frame shift, or a field carrying what the particles do not. That's why treat “closed system” as a verb, not a noun, and re-check it every time the numbers don’t add up. Do that, and momentum conservation stops being a trick and starts being a tool.
