1.4.4 · D5Momentum & Collisions
Question bank — System with external forces — conditions for conservation
The symbols you need (defined once, right here)
So nothing on this page uses a symbol you haven't met:
The master equation — pictured
Before you start, recall the one fact everything hangs on: Read it as: "the total momentum of the system only changes at the rate the net outside push dictates." Internal pushes never appear here — they cancel in equal-and-opposite pairs (Newton's Third Law).
Figure 1 makes this visual: internal arrows come in twins that cancel, so only the leftover outside arrow can move the group's momentum.
The three relaxations — defined and pictured
The parent note names three ways momentum can still be (partly) conserved even when an external force exists. Here is what each one actually means, so the Q&A below can use them freely:
True or false — justify
Total momentum is conserved whenever no force acts on any part of the system.
False as stated — what's required is that the net external force is zero; internal forces can be enormous (an explosion) yet total is untouched because they cancel in pairs.
If a nonzero external force acts, momentum cannot be conserved in any direction.
False — conservation is component-wise (relaxation D). If , then stays constant even while changes; a cannon fired horizontally conserves horizontal momentum despite gravity acting downward.
Momentum and kinetic energy are always conserved together.
False — they obey different conditions. needs ; kinetic energy needs the collision to be elastic. See Elastic vs Inelastic Collisions.
In a perfectly inelastic collision the objects stick together, so momentum is lost.
False — "sticking" loses kinetic energy, not momentum. As long as external impulse is negligible, total is fully conserved; the lost KE became heat/deformation.
Gravity is always an external force.
False — "external" depends on the boundary you draw. Gravity is external if the system is the ball alone, but internal if the system is ball + Earth. Same force, different label.
During a real bat–ball collision on Earth, momentum is exactly conserved.
False if you demand exactly; approximately true by relaxation I. Gravity acts, but over the millisecond contact its impulse is negligible, so we treat as conserved through the impact.
If total momentum is constant, then no external force acts.
True in the sense that the net external force must be zero — but individual external forces may exist and simply cancel (e.g. gravity balanced by a normal force on a horizontal table).
A system with zero total momentum can never have its momentum change.
False — zero is just a value. If a net external force acts, and the momentum leaves zero; the starting value doesn't protect it.
The center of mass of a system with moves in a straight line at constant velocity.
True — since (with the total mass, the center-of-mass acceleration), zero net external force means zero CM acceleration, so stays constant — the center of mass drifts uniformly regardless of internal chaos.
Spot the error
"A rocket in deep space speeds up, so its momentum increases — momentum isn't conserved for the rocket."
The error is choosing the wrong system. Take rocket + expelled exhaust: no external force, so total is conserved. The rocket alone isn't a closed system — the ejected gas carries the balancing momentum.
"The internal forces in an explosion are huge, so they must change the total momentum a lot."
Wrong — magnitude is irrelevant. By Newton's 3rd law each internal force has an equal-and-opposite twin; they cancel in the vector sum, so no internal force (however large) changes total .
"A ball bounces off the ground and reverses direction, so momentum is created from nothing."
The error is treating the ball alone as closed. The floor exerts an external normal impulse that reverses the ball's momentum; include ball + Earth and total is conserved (Earth recoils imperceptibly).
"Since and a force is present, must be changing — so total momentum can't be conserved."
The trap is the small-p/big-P slip: is the momentum of one particle, which a force does change. But total momentum obeys ; internal forces cancel and balanced external forces give zero net, so can stay constant even while individual change.
"Two cars crash, energy is lost to crumpling, so momentum is also lost by the same amount."
Momentum and energy are separate ledgers. Lost KE goes to deformation/heat, but momentum has no such sink — during the brief crash external impulse is negligible, so total is conserved even as KE drops.
"In a collision on a rough table, friction is external, so momentum is never conserved."
Over the short collision time friction's impulse is tiny compared with the huge contact impulse (relaxation I), so is approximately conserved through the impact. Friction matters only over the longer sliding phase afterward.
"A person jumps upward from the ground; their momentum increases, violating conservation."
For the person alone the ground's normal force is external, so their momentum legitimately changes. Take person + Earth and total is conserved — Earth acquires an equal downward momentum you can't perceive.
Why questions
Why do we bother checking the boundary before applying momentum conservation?
Because whether a force is internal or external — and thus whether is conserved — depends entirely on which objects are inside the system. The physics doesn't change, but the correct bookkeeping does.
Why is the correct test for the impulsive approximation a comparison of impulses, not forces?
Because what changes momentum is (relaxation A) — force times time, not force alone. A small force over a long time can beat a large force over a tiny time.
Why can internal forces never change total momentum, no matter how they're arranged?
Every internal force is paired with by Newton's Third Law; summing over all pairs gives exactly , so they drop out of — exactly the cancellation drawn in Figure 1.
Why is horizontal momentum conserved for a cannon firing over level ground, but vertical momentum is not?
The external forces (gravity down, normal up) act only vertically, so protects , while a nonzero vertical net force lets change — Newton's law holds independently per component (relaxation D).
Why does the statement "" use "if and only if"?
Because the implication runs both ways: zero net external force forces constant, and constant () forces the net external force to be zero. They are logically equivalent.
Why is "the same equation restated"?
Because (total mass times center-of-mass velocity), so differentiating gives . The master equation and the CM equation are two readings of one identity — see Center of Mass Motion.
Why can momentum be conserved in a collision even though there's a violent force between the bodies?
That violent force is internal to the two-body system, so it cancels in pairs; only the far weaker external forces (gravity, friction) could change , and their impulse over the short contact is negligible.
Edge cases
If the net external force is nonzero but perpendicular to the motion, what happens to momentum?
The momentum component along the motion is conserved, while the perpendicular component changes — conservation is decided axis by axis, not all-or-nothing (relaxation D).
A system's net external force is zero only for one instant. Is momentum conserved?
Only instantaneously — at that instant means is momentarily stationary, but if the force returns, resumes changing. Conservation over an interval needs the net force to stay zero throughout.
Consider a single free particle with no forces at all. Is its momentum "conserved"?
Yes, trivially — with its momentum is constant. A one-body system is the simplest case of Conservation of Linear Momentum.
Two external forces act on a system but exactly cancel. Is momentum conserved?
Yes — only the net external force enters . Two canceling externals (like gravity and a table's normal force) leave , so is conserved.
In the limit for a collision, why does even a large finite external force stop mattering?
Because its impulse as ; a finite force multiplied by a vanishing time contributes vanishing momentum change, so it drops out of the collision analysis (relaxation I).
A ball is dropped and caught by your hand — momentum clearly changes. Where did it go?
Into you and the Earth through your feet: the ball-alone system has an external upward impulse from your hand, but ball + hand + Earth conserves total , the balancing momentum flowing into the planet.
If external forces are present the whole time, is there any window where momentum conservation still applies?
Yes — during a sufficiently short sub-interval (a collision), the external impulse is negligible, so is conserved across that window even though it's not conserved over the full motion.
Connections
- Newton's Third Law — the reason internal-force pairs cancel.
- Impulse–Momentum Theorem — impulse, not force, is the right yardstick for the impulsive approximation.
- Center of Mass Motion — , the same law worn differently.
- Elastic vs Inelastic Collisions — the separate condition for kinetic energy.
- Conservation of Linear Momentum — the clean special case .