Constant vs Conserved: What Is the Difference?

A quantity is constant if its value does not change over time. It is conserved if it cannot be created or destroyed, only transferred. One does not imply the other: in a collision the momentum of either cart is conserved but not constant, because it moved rather than vanished.

AP Physics: Unit 4 (topics 3.4 Conservation of Energy, 4.1 Linear Momentum, 4.3 Conservation of Linear Momentum, 6.4 Conservation of Angular Momentum). College Board names this confusion itself. The appendix Vocabulary and Definitions of Important Ideas in AP Physics, printed at the back of all four Course and Exam Descriptions, carries a section headed Constant or conserved? that works the distinction through a leaking box of cereal and then through a falling box, and states that the choice of the system determines whether a quantity is constant or conserved. The framework then encodes the same distinction three times: LO 3.4.C, LO 4.3.B and LO 6.4.B are each worded as describing how the selection of a system determines whether that system's energy, momentum or angular momentum changes, and each is followed by three essential knowledge statements in matching order. EK 3.4.C.1, EK 4.3.B.1 and EK 6.4.B.1 state unconditionally that energy, momentum and angular momentum are conserved in all interactions. EK 3.4.C.2, EK 4.3.B.2 and EK 6.4.B.2 attach conditions before calling a system's total constant, and the third statement in each trio describes the failure case as transfer between the system and the environment. EK 3.4.B.3, EK 4.3.A.3.ii and EK 6.4.A.2.ii all begin with a system being selected so that a total is constant. The Topic 3.4 boundary statement adds that mechanical energy can be dissipated as thermal energy or sound by nonconservative forces, and the Unit 6 Preparing for the AP Exam guidance states that simply referencing a principle such as conservation of energy is not a complete enough answer to earn credit on the free-response section. Unit 3 carries 18 to 23 percent of the AP Physics 1 multiple-choice section, Unit 4 carries 10 to 15 percent and Unit 6 carries 5 to 8 percent.

College Board names this confusion, and then works it out

Almost every physics idea on this site is sourced from a course framework. This one is sourced from a section that exists only because students keep getting it wrong. At the back of all four Course and Exam Descriptions, in an appendix called Vocabulary and Definitions of Important Ideas in AP Physics, there is a heading that reads: Constant or conserved?

The appendix's framing sentence is the whole distinction: the cereal example is a good place to discuss the subtlety between the terms constant and conserved, and how the choice of the system determines whether a quantity is constant or conserved.

Here are the two ideas separated.

Constant is about time. A quantity is constant when its value at the end equals its value at the start, and at every moment in between. It is a claim about one number holding still. Ask it of a specific thing over a specific interval.

Conserved is about existence. A quantity is conserved when it cannot be created or destroyed. It can move: from one object to another, from one form to another, across the boundary of whatever you chose to call the system. What it cannot do is appear from nowhere or cease to exist. The appendix puts it plainly: the cereal does not simply vanish, disappear, or cease to exist because it is not selected to be part of the system. It is transferred out of the system.

So the sentence "momentum is conserved in this collision" and the sentence "momentum is constant in this collision" are different sentences. The first is true of momentum always. The second is true only of a particular system, and only if nothing outside it pushed.

Constant vs conserved, side by side

	Constant	Conserved
What it claims	The value does not change over time	The quantity cannot be created or destroyed
What it is a claim about	One quantity, in one system, over one interval	The quantity itself, everywhere, always
Depends on your choice of system	Yes, entirely	No
Depends on the time interval	Yes	No
Can the quantity move around inside the system	Yes, and the total still holds still	Yes, that is the whole idea
Can the quantity cross the system boundary	No. If it leaves, the total changed	Yes, and it is still conserved when it does
The CED's unconditional statements	None. Every one is conditional	Energy (EK 3.4.C.1), momentum (EK 4.3.B.1) and angular momentum (EK 6.4.B.1) are each conserved in all interactions
The CED's conditional statements	If the net external force on the selected system is zero, the total momentum of the system is constant (EK 4.3.B.2)	None
Applies to kinetic energy alone	Sometimes, as in uniform circular motion	No. Kinetic energy is not a conserved quantity; total energy is
Applies to one cart in a collision	No	Yes
What makes it false	Something crossed the boundary	Nothing in AP physics makes it false

Read the two rows about the CED's own statements together. College Board writes conserved with no conditions attached and constant with conditions attached every single time. That pattern is not accidental, and section six below shows it holding in three different units.

The case that separates them: a collision

Cart A, 0.40 kg, rolls at 2.5 m/s into cart B, 0.60 kg, at rest. They stick together. Track the momentum of three different systems through the same event.

System	Momentum before	Momentum after	Constant?	Conserved?
Cart A alone	$1.00\ \text{kg}\cdot\text{m/s}$	$0.40\ \text{kg}\cdot\text{m/s}$	No, it fell by 0.60	Yes
Cart B alone	$0$	$0.60\ \text{kg}\cdot\text{m/s}$	No, it rose by 0.60	Yes
Both carts	$1.00\ \text{kg}\cdot\text{m/s}$	$1.00\ \text{kg}\cdot\text{m/s}$	Yes	Yes

The two 0.60 figures in the middle column are the same 0.60. Cart A did not lose momentum in the sense of momentum going out of existence. It handed exactly that much to cart B, and EK 4.3.A.3 is the statement of it: in the absence of net external forces, any change to the momentum of an object within a system must be balanced by an equivalent and opposite change of momentum elsewhere within the system.

So momentum is conserved in all three rows and constant in only one of them. That single table is the reason the distinction matters. If conserved meant constant, the first two rows would be contradictions, and a student who believes they are contradictions concludes that momentum conservation does not apply to individual objects. It does. It applies to every one of them, which is precisely why the last row comes out even.

The same event, told in energy: the kinetic energy before is 1.25 J and after is 0.50 J, so 0.75 J of kinetic energy is not constant for any system here. Energy is still conserved, because that 0.75 J became thermal energy and sound. See what is conserved in a collision for the full accounting, and elastic vs inelastic collision for the classification.

The cereal box, which is where the appendix works it out

College Board's own example is worth following exactly, because it separates the two words using a quantity with no physics attached to it at all: pieces of cereal.

A box of cereal tears and cereal begins to spill. The physicist has a decision to make. System A continues to include every piece of cereal, even the pieces now on the floor. System B includes only the cereal still inside the box.

	System A: all the cereal	System B: cereal inside the box
Amount of cereal in the system	Does not change	Decreases as it spills
Constant?	Yes	No
Conserved?	Yes	Yes
Why	Nothing crossed the boundary you drew	Cereal crossed the boundary, and crossing is not vanishing

The appendix's own words on the second column are the ones to memorize: this cereal is still conserved, and the cereal does not simply vanish, disappear, or cease to exist because it is not selected to be part of the system. It is transferred out of the system.

Notice what changed between the two columns and what did not. The box is the same box. The tear is the same tear. The cereal on the floor is in exactly the same place either way. The only thing that changed is where a person drew a line, and the word constant flipped while the word conserved did not move. That asymmetry is the definition of the difference.

The appendix then repeats the exercise with energy. Let the box fall toward Earth. Choose the box and Earth together as the system, and the total mechanical energy is both conserved and constant: kinetic energy rises exactly as the gravitational potential energy of the box and Earth falls. Choose only the box, and the total energy of that system is not constant, because kinetic energy is arriving, and yet energy is still conserved. The appendix is careful about where the arriving energy came from: the increase is due to the transfer of energy into the box system by the external force of gravity doing work on the box, and that energy is not new energy created by gravity. The total energy of the universe has remained the same. See system and object vs system for how the boundary is chosen in the first place.

All four combinations exist, and you should be able to name one of each

Constant and conserved are independent properties, so there are four cases, and every one of them is a real situation in AP Physics.

	Conserved	Not conserved
Constant	Total momentum of two carts on a frictionless track that then collide. Total energy of a pendulum plus Earth with no air resistance.	Kinetic energy of a car in uniform circular motion at steady speed. Speed of a car on cruise control.
Not constant	Momentum of one cart during a collision. Energy of a falling box taken as the system on its own.	Kinetic energy of a block sliding to a halt on a rough floor. Mechanical energy of a bouncing ball.

The top right cell is the one people struggle to fill, so it is worth spelling out. A car going round a bend at a steady 18 m/s has a kinetic energy that never changes, so it is constant. Kinetic energy is nevertheless not a conserved quantity: nothing forbids it from being created out of chemical energy or destroyed into thermal energy, and it is created and destroyed constantly. Energy in total is conserved. Kinetic energy on its own is not, and the fact that a particular number happens to hold still does not promote it.

That car also fills the bottom left cell at the same instant. Its momentum has constant magnitude and a changing direction, so the momentum vector is not constant, while momentum remains a conserved quantity throughout. One vehicle, one moment, two of the four cells, which is the cleanest proof available that the two words are measuring different things.

What AP Physics 1 treats as conserved quantities: total energy, linear momentum, angular momentum, and electric charge. Kinetic energy, mechanical energy, potential energy and speed are not on that list. Mechanical energy in particular is often constant and is never described by the CED as conserved on its own: EK 3.4.C.2 says the total mechanical energy of the system is constant, and it attaches two conditions before it will say even that.

The CED encodes the distinction in its own learning objectives

This is the strongest evidence that the distinction is exam-relevant rather than pedantic. Three separate learning objectives in AP Physics 1, in three different units, are worded with the same shape, and each one is followed by three essential knowledge statements in the same order.

Energy, Topic 3.4. LO 3.4.C: describe how the selection of a system determines whether the energy of that system changes.

EK 3.4.C.1 Energy is conserved in all interactions.
EK 3.4.C.2 If the work done on a selected system is zero and there are no nonconservative interactions within the system, the total mechanical energy of the system is constant.
EK 3.4.C.3 If the work done on a selected system is nonzero, energy is transferred between the system and the environment.

Linear momentum, Topic 4.3. LO 4.3.B: describe how the selection of a system determines whether the momentum of that system changes.

EK 4.3.B.1 Momentum is conserved in all interactions.
EK 4.3.B.2 If the net external force on the selected system is zero, the total momentum of the system is constant.
EK 4.3.B.3 If the net external force on the selected system is nonzero, momentum is transferred between the system and the environment.

Angular momentum, Topic 6.4. LO 6.4.B: describe how the selection of a system determines whether the angular momentum of that system changes.

EK 6.4.B.1 Angular momentum is conserved in all interactions.
EK 6.4.B.2 If the net external torque exerted on a selected object or rigid system is zero, the total angular momentum of that system is constant.
EK 6.4.B.3 If the net external torque exerted on a selected object or rigid system is nonzero, angular momentum is transferred between the system and the environment.

Line them up and the pattern is unmistakable. Statement 1 in each trio uses conserved, applies to all interactions, and carries no conditions. Statement 2 uses constant, names the selected system, and carries a condition. Statement 3 describes what happens when the condition fails, and the word it reaches for is transferred, not lost. Those three learning objectives are the only ones in the AP Physics 1 framework worded as describing how the selection of a system determines whether a quantity changes, and the three quantities they name, energy, linear momentum and angular momentum, are three of the four the course treats as conserved. Electric charge is the fourth, and it is handled in AP Physics 2 without this wording.

Three more statements carry the same design. EK 3.4.B.3: a system may be selected so that the total energy of that system is constant. EK 4.3.A.3.ii: a system may be selected so that the total momentum of that system is constant. EK 6.4.A.2.ii: a system may be selected so that the total angular momentum of that system is constant. All three put the selecting first and the constancy second, in that order, because that is the order the physics runs in.

The conditions differ from quantity to quantity, and mixing them up is its own error. Linear momentum's condition is about external forces; angular momentum's is about external torques; energy's is about external work and about what happens inside, because EK 3.4.C.2 also requires no nonconservative interactions within the system. A block sliding to a stop on a rough floor, with the floor and Earth inside the system, has constant momentum and falling mechanical energy at the same time, because friction is internal and nonconservative. Isolated system sets out each condition.

Where it costs a mark

Every item here is a scoring event on a free-response question, not a matter of taste.

Writing "momentum is conserved" with no system named. EK 4.3.B.2 makes the useful claim conditional on the net external force on the selected system. A justification that never selects a system has not stated the condition, so it has not made the argument. Name the system in the same sentence.
Concluding that a single object's momentum cannot change because momentum is conserved. It changes constantly. Conservation says the change was matched somewhere else, not that it did not happen.
Saying energy was lost. EK 3.4.C.1 says energy is conserved in all interactions, so nothing was lost. It was transferred or converted. The Topic 3.4 boundary statement names the usual destination: mechanical energy can be dissipated as thermal energy or sound by nonconservative forces. Write "converted to thermal energy", not "lost".
Saying momentum is not conserved in an inelastic collision. It is. Momentum is conserved in every collision. What is not constant in an inelastic collision is the system's kinetic energy, which is a different quantity and a different word.
Calling mechanical energy conserved when friction acts. Mechanical energy is not on the list of conserved quantities in the first place, and with friction inside the system it is not even constant. Total energy is conserved either way.
Treating "the total energy of the universe is constant" as the useful statement. It is true and it solves nothing, because you cannot write an equation for the universe. The usable statement is always about a chosen system over a chosen interval.
Switching systems mid-answer. If part (a) chose the two carts and part (b) quietly uses one cart, the conditions you cited in (a) no longer hold in (b), and the marker follows what you wrote.
Assuming constant implies conserved. A car at steady speed on a bend has constant kinetic energy, and kinetic energy is not a conserved quantity. Constancy of a number is not a conservation law.

How to say it, and what the exam will and will not ask

College Board says outright that naming the principle is not enough. The Unit 6 guidance under Preparing for the AP Exam warns that when writing justifications for claims, simply referencing an equation, law, or physical principle is not sufficient, and its worked example of a failure is precisely this: stating that one disk is rolling faster than another because of conservation of energy is not a complete enough answer to earn credit on the free-response section of the exam. Students must clearly and concisely explain the steps in their reasoning that lead from the equation, law, or physical principle to the justification of their claim.

Which is why the wording matters. A sentence template that survives scrutiny: "Taking [these objects] as the system, the net external force on it is [zero / not zero] during [this interval], so the system's total momentum is [constant / not constant]. Momentum is conserved throughout; the change in cart A's momentum is matched by an equal and opposite change in cart B's." Every clause in that sentence is doing work, and each one maps onto an essential knowledge statement.

Two pieces of scaffolding make the claim safe in practice.

First, the collision model itself. EK 4.1.A.3.i defines a collision as an interaction where the forces exerted between the involved objects in the system are much larger than the net external force exerted on those objects during the interaction. That is what lets you call the two-cart system's momentum constant across a collision on a real bench with real friction and real gravity: the external forces are there, they are just too small to matter over a few milliseconds. The word during is load bearing, and a question that asks about the second after the collision is asking about a different interval.

Second, the appendix's reassurance. The AP Physics Exam will not directly assess student understanding of physics vocabulary, and it says so explicitly: students will not be asked to identify the correct definition of acceleration, or the difference between a system and an object. So no question will ask you to define constant and conserved. Every question that touches momentum or energy conservation will assess whether you can use them correctly, which is why the distinction is worth learning and why memorizing the two definitions on their own is not enough.

Where this sits on the exam: Topic 3.4 Conservation of Energy in Unit 3, which is 18 to 23 percent of the AP Physics 1 multiple-choice section, and Topic 4.3 Conservation of Linear Momentum in Unit 4, which is 10 to 15 percent. The equation sheet prints $\vec{J} = \vec{F}_{\text{avg}} \Delta t = \Delta \vec{p}$ and $\vec{v}_{\text{cm}} = \frac{\sum \vec{p}_i}{\sum m_i}$ but no line that says the words conserved or constant, because those words are conditions on a system rather than formulas. The full sheet is here.

A collision where momentum is conserved and constant for nobody in particular

Cart A has mass 0.40 kg and moves right at 2.5 m/s on a level frictionless track. It collides with cart B, mass 0.60 kg, at rest, and the two stick together. The carts are in contact for 0.050 s. Find the final velocity, the change in each cart's momentum, the average force between them, and state for each of the three possible systems whether momentum is constant and whether it is conserved.

Declare the convention: positive is to the right, and all momenta are components along that axis.
Choose the system for the calculation: both carts together. The track is level and frictionless, so the vertical forces cancel and there is no external horizontal force. EK 4.3.B.2 then makes the total momentum constant.
Total momentum before: $p_0 = (0.40\ \text{kg})(2.5\ \text{m/s}) + (0.60\ \text{kg})(0) = 1.00\ \text{kg}\cdot\text{m/s}$ .
They stick, so afterwards one combined mass of 1.00 kg moves at a single velocity $v$ . Setting the total equal: $1.00 = (1.00\ \text{kg})v$ , so $v = 1.00\ \text{m/s}$ to the right.
Cart A on its own: $p_{A0} = 1.00$ and $p_{Af} = (0.40)(1.00) = 0.40\ \text{kg}\cdot\text{m/s}$ , so $\Delta p_A = 0.40 - 1.00 = -0.60\ \text{kg}\cdot\text{m/s}$ . Cart A's momentum is not constant.
Cart B on its own: $p_{B0} = 0$ and $p_{Bf} = (0.60)(1.00) = 0.60\ \text{kg}\cdot\text{m/s}$ , so $\Delta p_B = +0.60\ \text{kg}\cdot\text{m/s}$ . Cart B's momentum is not constant either.
Add them: $\Delta p_A + \Delta p_B = -0.60 + 0.60 = 0$ . That is EK 4.3.A.3 in numbers, and it is why the total came out constant.
Average force from the impulse-momentum theorem applied to cart A: $F_{\text{avg}} = \Delta p_A / \Delta t = (-0.60\ \text{kg}\cdot\text{m/s})/(0.050\ \text{s}) = -12\ \text{N}$ , so 12 N directed left. By Newton's third law cart A pushes B rightward with 12 N, giving B an impulse of $(+12\ \text{N})(0.050\ \text{s}) = +0.60\ \text{kg}\cdot\text{m/s}$ , which matches $\Delta p_B$ exactly.
Kinetic energy, to show a second quantity behaving differently: before, $K_0 = \frac{1}{2}(0.40)(2.5)^2 = \frac{1}{2}(0.40)(6.25) = 1.25\ \text{J}$ ; after, $K_f = \frac{1}{2}(1.00)(1.00)^2 = 0.50\ \text{J}$ . The system's kinetic energy fell by 0.75 J.

Final velocity 1.00 m/s right, $\Delta p_A = -0.60$ and $\Delta p_B = +0.60\ \text{kg}\cdot\text{m/s}$ , average force 12 N. Momentum is conserved for all three systems and constant only for the two-cart system. The 0.75 J of kinetic energy that stopped existing as kinetic energy did not stop existing: it became thermal energy and sound, so total energy is conserved as well, while neither the kinetic energy nor the mechanical energy of this system is constant.

The appendix's falling box, with numbers on both choices of system

A 0.45 kg box of cereal falls from rest through 1.8 m. Ignore air resistance and use $g = 9.8\ \text{m/s}^2$ . Analyze it twice, once with the box alone as the system and once with the box and Earth together, and say in each case whether the system's total energy is constant and whether energy is conserved.

Declare the convention: take downward displacement as positive for the fall, and set the gravitational potential energy of the box and Earth to zero at the landing point.
Find the speed at the bottom from kinematics: $v^2 = v_0^2 + 2a\Delta y = 0 + 2(9.8\ \text{m/s}^2)(1.8\ \text{m}) = 35.28\ \text{m}^2/\text{s}^2$ , so $v = \sqrt{35.28} = 5.94\ \text{m/s}$ .
Kinetic energy at the bottom: $K = \frac{1}{2}(0.45\ \text{kg})(35.28\ \text{m}^2/\text{s}^2) = 7.94\ \text{J}$ . It started at zero.
System 1, the box alone. Its total energy went from 0 J to 7.94 J, so it is not constant. Gravity is exerted by Earth, which is outside this boundary, so it is an external force and it does work on the box: $W = mg\Delta y = (0.45)(9.8)(1.8) = 7.94\ \text{J}$ , matching the kinetic energy gained.
Apply EK 3.4.C.3 to that: the work done on the selected system is nonzero, so energy is transferred between the system and the environment. Transferred in, in this case. Energy is still conserved, because the 7.94 J came from somewhere rather than being created by gravity.
System 2, the box and Earth together. Gravity is now an interaction between two members of the system, so it is internal, and instead of doing external work it changes a potential energy the system owns: $\Delta U_g = mg\Delta y = -7.94\ \text{J}$ over the fall, since the box moves downward.
Add the two changes for system 2: $\Delta K + \Delta U_g = +7.94 - 7.94 = 0\ \text{J}$ . The total mechanical energy of the box and Earth system is constant, which is EK 3.4.C.2 with its two conditions met: no external work, no nonconservative interaction inside.
Check that both descriptions give the same physical prediction. Both say the box arrives at 5.94 m/s carrying 7.94 J of kinetic energy. They disagree about bookkeeping, not about the world.

Box alone: total energy not constant (0 J to 7.94 J), energy conserved. Box and Earth: total mechanical energy constant at whatever value you assigned it, energy conserved. Speed at the bottom is 5.94 m/s either way. This is the appendix's second example made numeric, and its warning applies to the 7.94 J: the energy transferred into the box by the force of gravity is not new energy created by gravity. Moving the boundary moved the 7.94 J from the label "work done on the system" to the label "potential energy inside the system", and that is the only thing that moved.

A block sliding to a halt, where nothing about it is constant and everything is conserved

A 2.0 kg block slides across a level floor at 3.5 m/s and is brought to rest by friction, with a coefficient of kinetic friction of 0.30. Find the stopping distance, the stopping time, the friction impulse and the thermal energy generated. Then state which quantities are constant and which are conserved for the block alone and for the block, floor and Earth together.

Declare the convention: positive is the direction of motion, and the floor is horizontal so the normal force is $F_N = mg = (2.0\ \text{kg})(9.8\ \text{m/s}^2) = 19.6\ \text{N}$ .
Friction force: $F_f = \mu_k F_N = (0.30)(19.6\ \text{N}) = 5.88\ \text{N}$ , directed backward.
Acceleration: $a = -F_f/m = -(5.88\ \text{N})/(2.0\ \text{kg}) = -2.94\ \text{m/s}^2$ .
Stopping distance from $v^2 = v_0^2 + 2ad$ with $v = 0$ : $d = -v_0^2/(2a) = (3.5)^2/(2 \times 2.94) = 12.25/5.88 = 2.08\ \text{m}$ .
Stopping time from $v = v_0 + at$ : $t = v_0/2.94 = 3.5/2.94 = 1.19\ \text{s}$ .
Friction impulse on the block: $J = F_f t = (5.88\ \text{N})(1.19\ \text{s}) = 7.0\ \text{N}\cdot\text{s}$ backward, which matches the block's momentum change from $(2.0)(3.5) = 7.0\ \text{kg}\cdot\text{m/s}$ down to zero.
Thermal energy generated: $F_f d = (5.88\ \text{N})(2.08\ \text{m}) = 12.25\ \text{J}$ , equal to the block's initial kinetic energy $\frac{1}{2}(2.0)(3.5)^2 = \frac{1}{2}(2.0)(12.25) = 12.25\ \text{J}$ . The two agree to the digit, which is the check.
Block alone as the system: momentum falls from 7.0 to 0 and kinetic energy falls from 12.25 J to 0, so nothing about it is constant. Friction and the normal force are external, so EK 4.3.B.3 and EK 3.4.C.3 both apply: momentum and energy are transferred out.
Block, floor and Earth as the system: friction is now internal, so the total momentum is constant, and the 7.0 kg m/s the block gave up went into Earth, whose enormous mass makes the resulting speed unmeasurable but not zero. The total energy is also constant, with the 12.25 J now sitting as thermal energy in the block and the floor rather than as kinetic energy.
Note the asymmetry between the two conservation conditions. In the second system the momentum is constant and the mechanical energy is not, because EK 3.4.C.2 requires no nonconservative interactions inside the system, and friction is exactly that.

Stopping distance 2.08 m, stopping time 1.19 s, friction impulse 7.0 N s backward, thermal energy 12.25 J. For the block alone nothing is constant, and both momentum and energy are still conserved. For the block, floor and Earth, momentum and total energy are both constant while mechanical energy is not. A quantity that is conserved can go to zero for the thing you are looking at; that is what transferred means.

Frequently asked questions

What is the difference between constant and conserved in physics?

Constant means a quantity's value does not change over time. Conserved means the quantity cannot be created or destroyed, only transferred from place to place or from one form to another. They are independent claims. College Board's own appendix, Vocabulary and Definitions of Important Ideas in AP Physics, has a section headed Constant or conserved? that separates them using a leaking box of cereal: if you count only the cereal inside the box, the amount is not constant, because cereal is spilling out, but the cereal is still conserved, because it does not vanish or cease to exist just because you did not include it in your system. Constant depends on where you drew the system boundary. Conserved does not.

Is momentum conserved or constant in a collision?

Both, but of different things. Momentum is conserved in all interactions, which AP Physics 1 states unconditionally in EK 4.3.B.1. The total momentum of a chosen system is constant only when the net external force on that system is zero, which is EK 4.3.B.2. In a two-cart collision on a frictionless track, the total momentum of the two carts together is constant, while the momentum of each individual cart changes a great deal. Cart A's loss equals cart B's gain to the digit. Saying momentum is conserved for one cart is correct; saying it is constant for one cart is wrong, and getting that backwards is the error the distinction exists to prevent.

Can something be conserved but not constant?

Yes, and it is the case that comes up constantly in mechanics. The momentum of a single cart in a collision is conserved and not constant: it changed, but it changed by handing an equal amount to the other cart rather than by disappearing. The energy of a falling box, taken as a system on its own, is conserved and not constant: kinetic energy is arriving from outside the boundary as gravity does work. The appendix words the general case as transfer. A conserved quantity is always allowed to leave the system you selected, and when it does, the total inside your boundary changes while the conservation law remains completely intact.

Can something be constant but not conserved?

Yes. A car rounding a bend at a steady 18 m/s has a kinetic energy that never changes, so its kinetic energy is constant. Kinetic energy is nevertheless not a conserved quantity: it is created out of chemical energy every time an engine runs and destroyed into thermal energy every time a brake is applied. Speed, temperature and mechanical energy behave the same way, holding steady in particular situations without obeying a conservation law. The quantities AP Physics treats as genuinely conserved are total energy, linear momentum, angular momentum and electric charge. A number holding still is not evidence that it belongs on that list.

Why does the choice of system decide whether a quantity is constant?

Because the boundary decides which interactions count as external, and only external ones can move a quantity in or out. AP Physics 1 EK 2.3.A.2 says interactions between objects within a system, the internal forces, do not influence the motion of the system's center of mass, so anything that happens entirely inside the boundary just shuffles the quantity around without changing the total. Cross the boundary and the total changes. That is why EK 4.3.B.2 and EK 3.4.C.2 both begin by naming the selected system before making any claim about constancy. Redrawing the boundary changes the arithmetic without changing a single thing about the physical situation.

Is energy lost to friction?

No, and writing that it is will cost you a justification mark. AP Physics 1 EK 3.4.C.1 states that energy is conserved in all interactions, so no energy is ever lost anywhere in the course. What happens with friction is a conversion: the Topic 3.4 boundary statement says students are expected to know that mechanical energy can be dissipated as thermal energy or sound by nonconservative forces. The mechanical energy of the system falls, which is a true and useful statement, and the total energy does not change. The safe phrasings are converted to thermal energy, dissipated as thermal energy, or transferred out of the system.

Does conserved mean the same thing as constant if the system is isolated?

In that special case the two statements happen to be true together, which is exactly why the confusion survives. An isolated system has nothing crossing its boundary, so a conserved quantity inside it cannot go anywhere and its total is also constant. But the two sentences still mean different things, and only one of them stops being true when you change the system. Move the boundary so that something crosses it and constant fails immediately while conserved is untouched. Learning the distinction on the isolated case alone is how students arrive at the belief that an individual object's momentum cannot change.