AP Physics 1 · Topic 3.4

Topic 3.4: Conservation of Energy

Unit 3: Work, Energy, and Power18-23% of the multiple-choice section

Energy is conserved in all interactions. Mechanical energy, the sum of a system's kinetic and potential energies, is constant only when the work done on the system is zero and no nonconservative interaction acts inside it. Choose the system first: that choice decides what you track.

AP Physics: Unit 3 (topics 3.4 Conservation of Energy). AP Physics 1 Unit 3, Topic 3.4, which has three learning objectives: 3.4.A describe the energies present in a system, 3.4.B describe the behavior of a system using conservation of mechanical energy principles, and 3.4.C describe how the selection of a system determines whether the energy of that system changes. Its nine essential knowledge statements establish that a single-object system can only have kinetic energy, that mechanical energy is the sum of a system's kinetic and potential energies, that energy is conserved in all interactions, and that mechanical energy is constant only when the work done on the system is zero and no nonconservative interaction acts inside it. The boundary statement adds that mechanical energy can be dissipated as thermal energy or sound by nonconservative forces. Unit 3 carries 18 to 23 percent of the multiple-choice section and about 22 to 27 class periods, and the suggested skills for this topic are 1.A, 2.A, 2.C, and 3.C.

What Topic 3.4 requires

Topic 3.4 is the only topic in Unit 3 with three learning objectives. The other four topics have one each.

3.4.A Describe the energies present in a system.
3.4.B Describe the behavior of a system using conservation of mechanical energy principles.
3.4.C Describe how the selection of a system determines whether the energy of that system changes.

Nine essential knowledge statements sit under those three, and not one of them is an equation.

3.4.A.1 A system composed of only a single object can only have kinetic energy.
3.4.A.2 A system that contains objects that interact via conservative forces or that can change its shape reversibly may have both kinetic and potential energies.
3.4.B.1 Mechanical energy is the sum of a system's kinetic and potential energies.
3.4.B.2 Any change to a type of energy within a system must be balanced by an equivalent change of other types of energies within the system or by a transfer of energy between the system and its surroundings.
3.4.B.3 A system may be selected so that the total energy of that system is constant.
3.4.B.4 If the total energy of a system changes, that change will be equivalent to the energy transferred into or out of the system.
3.4.C.1 Energy is conserved in all interactions.
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.
3.4.C.3 If the work done on a selected system is nonzero, energy is transferred between the system and the environment.

One boundary statement closes the topic: AP Physics 1 expects students to know that mechanical energy can be dissipated as thermal energy or sound by nonconservative forces.

The suggested skills are 1.A, create diagrams, tables, charts, or schematics to represent physical situations; 2.A, derive a symbolic expression from known quantities by selecting and following a logical mathematical pathway; 2.C, compare physical quantities between two or more scenarios or at different times and locations in a single scenario; and 3.C, justify or support a claim using evidence from experimental data, physical representations, or physical principles or laws. None of the four is the calculation skill 2.B: this topic is assessed by diagram, derivation, comparison, and justification. Unit 3 carries 18 to 23 percent of the multiple-choice section and about 22 to 27 class periods.

The system is the first decision (3.4.A.1 and 3.4.A.2)

Before you can say what energy a system has, you have to say what the system is. Learning objective 3.4.C makes that explicit: describe how the selection of a system determines whether the energy of that system changes.

Draw an imaginary boundary around whatever you want to call the system. Everything inside is yours to track. Everything outside is the environment, and it can only reach you by doing work.

One object inside the boundary. Statement 3.4.A.1 is blunt: a system composed of only a single object can only have kinetic energy. There is nothing else for it to have, because potential energy needs an interacting pair, which is Topic 3.3 essential knowledge 3.3.A.1. Gravity is then an external force, and it does work across the boundary.
Two or more interacting objects inside the boundary. Statement 3.4.A.2 says a system that contains objects that interact via conservative forces or that can change its shape reversibly may have both kinetic and potential energies. Put the Earth inside the boundary with a falling rock and gravity is no longer external. It becomes an internal interaction with a potential energy attached, and no work crosses the boundary at all.

The second half of 3.4.A.2 is easy to skim past. Or that can change its shape reversibly is how a spring gets in. A system containing a compressed spring has elastic potential energy because that deformation is reversible. A crumpled can does not, because it is not.

Both boundaries are legal, and they answer any physical question identically. The first worked example below runs the same falling rock both ways and gets the same speed twice. What you cannot do is mix them: count gravity as external work and as a potential energy in one equation and you have banked the same energy twice. That is the accounting error the work-energy theorem guide and the conservation of energy guide both warn about, from opposite sides.

Mechanical energy, and the exact condition for it to hold (3.4.B.1, 3.4.C.2)

Mechanical energy is the sum of a system's kinetic and potential energies. That is the whole of 3.4.B.1, and the word sum is doing real work: thermal energy and sound are not in it.

E_{\text{mech}} = K + U

Write that symbol freely: the CED uses it too, and essential knowledge 3.2.A.4.iii writes the friction loss as $\Delta E_{\text{mech}} = F_f d\cos\theta$ . What no page of the equation sheet carries is a balance built from it. The AP Physics 1 equation sheet prints no conservation of energy equation at all. What it does print is every piece you would build one from: $K = \frac{1}{2}mv^2$ , $W = F_{\parallel}d = Fd\cos\theta$ , the work-energy theorem $\Delta K = \sum W_i = \sum F_{\parallel,i} d_i$ , $U_s = \frac{1}{2}k(\Delta x)^2$ , $U_G = -\frac{Gm_1m_2}{r}$ , and $\Delta U_g = mg\Delta y$ . It leaves the balance for you to assemble. Scanning the AP Physics 1 formula sheet for an energy conservation line and not finding one is not a fault in the sheet.

Statement 3.4.C.2 supplies the condition, and it is a conjunction. 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. Both clauses have to hold.

Zero work done on the system. Nothing outside the boundary is pushing energy in or pulling it out. A hand shoving, a rope hauling, or a motor driving all break this clause.
No nonconservative interactions inside. Friction and air resistance are the examples the CED names, in 3.2.A.1.v. Anything inside your boundary that turns mechanical energy into thermal energy or sound breaks this clause.

Fail either one and mechanical energy is not constant. It does not follow that energy is not conserved, which is the next section. Notice too that both clauses are statements about the boundary you drew, so you can sometimes rescue the first by redrawing. A rope running over a pulley to a hanging block does external work on a cart if the system is the cart alone, and does no external work at all if the system is cart, rope, block, and Earth together.

Energy is conserved in all interactions (3.4.C.1)

Statement 3.4.C.1 is one sentence with no conditions attached: energy is conserved in all interactions. Conservation of mechanical energy, by contrast, arrives with the two clauses above. Keeping those two claims apart is most of Topic 3.4.

Claim	When it holds
Energy is conserved in all interactions	Always. 3.4.C.1 attaches no condition.
The total energy of a chosen system is constant	When no energy crosses the boundary, which is what 3.4.B.3 and 3.4.B.4 describe.
The mechanical energy of a chosen system is constant	Only when the work done on the system is zero and no nonconservative interaction acts inside it (3.4.C.2).

So when a block skids to a stop, nothing about 3.4.C.1 is in danger. The kinetic energy did not vanish. It turned into a form that is not mechanical, and the topic's boundary statement names the destinations: AP Physics 1 expects students to know that mechanical energy can be dissipated as thermal energy or sound by nonconservative forces.

That gives you the sentence to write on a free response. Energy was lost is vague, and skill 3.C asks you to justify a claim using physical principles. The block's kinetic energy was dissipated as thermal energy by friction between the block and the floor, so the mechanical energy of the block and floor system decreased while the total energy did not change is the same idea with the accounting on show.

For the size of the loss, Topic 3.2 supplies a working rule in 3.2.A.4.iii: the energy dissipated by friction is typically equated to the force of friction times the length of the path over which the force is exerted. Read length of the path, not displacement. A block pushed across a rough floor and back finishes with zero displacement and pays friction for the whole round trip.

Crossing the boundary: transfer in, transfer out (3.4.B.2, 3.4.B.4, 3.4.C.3)

Three statements describe what happens when the boundary is not sealed, and together they form a bookkeeping identity.

3.4.B.2 Any change to a type of energy within a system must be balanced by an equivalent change of other types of energies within the system or by a transfer of energy between the system and its surroundings.
3.4.B.4 If the total energy of a system changes, that change will be equivalent to the energy transferred into or out of the system.
3.4.C.3 If the work done on a selected system is nonzero, energy is transferred between the system and the environment.

Read as one sentence: energy that leaves one store either shows up in another store inside the same boundary or it crossed the boundary. There is no third option and no leak.

Work is the transfer mechanism this course gives you. Topic 3.2 defines work as the amount of energy transferred into or out of a system by a force exerted on that system over a distance (3.2.A.1), and 3.4.C.3 closes the loop from the other end. Positive external work raises the system's total energy by exactly that amount; negative external work lowers it by exactly that amount. The third worked example below is that statement with numbers in it.

One piece of vocabulary you will meet elsewhere and not here: the AP Physics 1 course description never calls a system open, closed, or isolated. It states the same idea as a decision, in 3.4.B.3, that a system may be selected so that the total energy of that system is constant. If a textbook hands you closed system, translate it into the CED's terms as a boundary chosen so that no energy crosses it, which by 3.4.B.4 means the total energy inside does not change.

Why the zero of potential energy is yours to choose

Every energy balance with a height in it needs a zero for gravitational potential energy, and Topic 3.3 essential knowledge 3.3.A.3 says whose decision that is: the definition of zero potential energy for a given system is a decision made by the observer considering the situation to simplify or otherwise assist in analysis.

That freedom never threatens the conservation statement, and the algebra shows why. Suppose you and a classmate pick zeros that differ, so every potential energy you write differs from theirs by the same constant $C$ . Your balance reads

K_i + U_i + C = K_f + U_f + C

and the $C$ cancels. You disagree about every individual value of $U$ and you agree exactly about $\Delta U$ , and only $\Delta U$ ever enters the physics. The sheet's other gravitational form, $U_G = -\frac{Gm_1m_2}{r}$ , is an absolute value only because it fixes the zero at infinite separation for you. That is also why the equation sheet prints the near-surface relation as a change, $\Delta U_g = mg\Delta y$ , rather than as a bare $U = mgh$ .

Two habits keep it clean. Pick the zero before you write your first equation, usually at the lowest point in the problem so every gravitational term stays positive. Then hold it. Moving the zero halfway through a solution does the same damage as flipping the positive direction of an axis halfway through, and it is harder to spot afterwards.

The same goes for the boundary. Name the system in the first line, write down the zero and the positive direction beside it, and let the rest of the solution follow. Skill 2.A for this topic is about deriving a symbolic expression by following a logical pathway, and a pathway that starts with three declared choices is a pathway a grader can follow.

Work and heating: what Topic 3.4 covers and what it defers

Unit 3 draws a hard edge around how far the energy accounting goes, and it draws it in the Topic 3.2 boundary statement rather than in Topic 3.4. It is worth quoting whole, because the last sentence is the part that gets dropped:

AP Physics 1 only expects students to analyze the transfer of mechanical energy (as defined in Unit 3, Topic 4: Conservation of Energy), although students should be aware that mechanical energy may be dissipated in the form of thermal energy or sound. In AP Physics 2, students will also study how thermal energy can be transferred between systems through heating or cooling.

Three consequences follow.

Work is the transfer you compute. Every energy crossing a boundary in AP Physics 1 does so as work done by a force. That is 3.4.C.3.
Dissipation is a destination you name, not a temperature you calculate. You are expected to say that mechanical energy became thermal energy or sound and to carry the loss in the balance. You are not expected to compute a temperature change: $Q = mc\Delta T$ is printed on the AP Physics 2 equation sheet, not the AP Physics 1 one.
Heating is out of scope here. The word appears once in the whole AP Physics 1 course description, in the sentence above, and only to hand the subject to AP Physics 2. A Physics 1 question that mentions a rough surface wants a dissipation term, not a thermodynamics problem.

In AP Physics 2 the same bookkeeping reappears as the first law of thermodynamics, with heating added to work as a second way for energy to cross a boundary; the PV diagrams guide picks the story up there. The principle does not change. The list of transfer mechanisms gets one entry longer.

Representing the ledger, and where Topic 3.4 leads

Suggested skill 1.A for this topic asks you to create diagrams, tables, charts, or schematics to represent physical situations, and for energy the standard representation is the bar chart: one bar per store, one chart per snapshot, drawn to scale.

Bar charts make two statements visible at a glance. Because 3.4.B.1 defines mechanical energy as a sum, the total height of the bars is the quantity you are claiming stays the same. Because 3.4.B.2 says any change in one store must be balanced elsewhere, a bar that shrinks between snapshots forces either a bar that grows or an arrow crossing the boundary.

The CED's own optional activities for this topic run on that idea. A four-square activity listed for Topics 3.2 and 3.4 has students describe an everyday situation in words, draw a free-body diagram with the displacement beside it, draw initial and final energy bar charts, and then state for each force in the diagram whether it does positive or negative work and what energy transformation that force is responsible for. Another asks students to explain, using energy and circular motion principles, why a cart released from rest must start higher than the top of a vertical loop to get around it.

From here the unit has one topic left. Topic 3.5 takes this same ledger and divides it by time, which is all power is. Beyond Unit 3, Unit 4 brings the course's second conservation law, and conservation of linear momentum reads well against this page: momentum is conserved under a condition on external forces, energy under a condition on external work, and elastic and inelastic collisions are sorted by whether kinetic energy survives the interaction. For the step-by-step solving routine, use the conservation of energy guide.

One falling rock, two system boundaries

A 1.5 kg rock is released from rest and falls 4.0 m. Air resistance is negligible. Find its speed just before it lands, first taking the system to be the rock alone, then taking it to be the rock and the Earth together.

Declare the conventions before anything else. Up is positive, and the zero of gravitational potential energy sits at the landing point. The rock falls 4.0 m, so $\Delta y = -4.0 \, \mathrm{m}$ , and with $g = 9.8 \, \mathrm{m/s^2}$ the useful product is $mg = (1.5)(9.8) = 14.7 \, \mathrm{N}$ .
System: the rock alone. By 3.4.A.1 this system can only have kinetic energy, so there is no $U$ term to write. Gravity is external and does work across the boundary: $W = F_{\parallel}d = (14.7 \, \mathrm{N})(4.0 \, \mathrm{m}) = 58.8 \, \mathrm{J}$ , positive because the force and the displacement both point down. The work done on the system is nonzero, so by 3.4.C.3 energy is transferred in, and by 3.4.B.4 the system's total energy rises by exactly that much.
Apply the sheet's work-energy theorem, $\Delta K = \sum W_i$ , with $K_i = 0$ . Then $K_f = 58.8 \, \mathrm{J}$ and $v = \sqrt{2K_f/m} = \sqrt{2(58.8)/1.5} = \sqrt{78.4} = 8.85 \, \mathrm{m/s}$ .
System: the rock and the Earth. Now gravity is internal, so by 3.4.A.2 this system has a gravitational potential energy. Nothing outside does work on it and nothing inside it is nonconservative, so both clauses of 3.4.C.2 hold and the mechanical energy is constant.
Write that as $\Delta K + \Delta U_g = 0$ . With $\Delta U_g = mg\Delta y = (14.7)(-4.0) = -58.8 \, \mathrm{J}$ , the kinetic energy must rise by $58.8 \, \mathrm{J}$ , giving $K_f = 58.8 \, \mathrm{J}$ and the same $v = 8.85 \, \mathrm{m/s}$ .
Compare the two ledgers. The same 58.8 J appears once as external work and once as a drop in an internal potential energy. It never appears twice in one equation, which is the error to watch for.

$v = 8.9 \, \mathrm{m/s}$ to two significant figures, from either system. The rock-alone system gained energy from outside; the rock-and-Earth system kept its mechanical energy constant and moved it between stores. Both descriptions are correct, and learning objective 3.4.C is the statement that they have to agree.

A rough ramp: mechanical energy falls, total energy does not

A 3.0 kg block starts from rest and slides 2.0 m down a ramp inclined at $30^\circ$ to the horizontal. A constant 4.5 N friction force acts on it along the slope. Take the system to be the block, the ramp, and the Earth. Find the block's speed at the end of the 2.0 m run, and account for every joule.

Set the conventions and the boundary. The positive direction of motion is down the slope; the vertical coordinate is measured upward with its zero at the end of the run. Inside the boundary, gravity is a conservative interaction between block and Earth, and friction between block and ramp is a nonconservative interaction. Nothing outside does work on this system, so the first clause of 3.4.C.2 holds and the second fails: the total energy is constant, the mechanical energy is not.
Find the height change. The run is 2.0 m along a slope at $30^\circ$ , so $\Delta y = -(2.0)\sin 30^\circ = -1.0 \, \mathrm{m}$ .
Gravitational term: $\Delta U_g = mg\Delta y = (3.0)(9.8)(-1.0) = -29.4 \, \mathrm{J}$ . The gravitational store gave up 29.4 J.
Dissipation, using the Topic 3.2 rule that the energy dissipated by friction is equated to the friction force times the length of the path over which it acts: $\Delta E_{\text{thermal}} = (4.5 \, \mathrm{N})(2.0 \, \mathrm{m}) = 9.0 \, \mathrm{J}$ . The path length is a positive distance, which is why this term is not signed by the vertical axis.
Balance the books with 3.4.B.2. The 29.4 J that left the gravitational store had to reappear inside the boundary. 9.0 J of it became thermal energy, so $K_f = 29.4 - 9.0 = 20.4 \, \mathrm{J}$ .
Speed: $v = \sqrt{2K_f/m} = \sqrt{2(20.4)/3.0} = \sqrt{13.6} = 3.69 \, \mathrm{m/s}$ .
Check the two conservation claims separately. Mechanical energy went from 29.4 J of potential (with $K_i = 0$ ) to 20.4 J of kinetic, a fall of 9.0 J. Total energy went from 29.4 J to 20.4 J of kinetic plus 9.0 J of thermal, which is 29.4 J again. Nothing was lost.

The block reaches $3.7 \, \mathrm{m/s}$ . The system's mechanical energy fell by 9.0 J, dissipated by the nonconservative friction interaction into the forms the Topic 3.4 boundary statement names, thermal energy or sound. Total energy did not change at all, so 3.4.C.1 is untouched while 3.4.C.2 simply does not apply here.

Work from outside changes the system's total energy

A 2.5 kg crate is pulled from rest up a frictionless slope by a rope. Over the pull, the rope does 120 J of work on the crate, and the crate rises 3.0 m vertically. Taking the system to be the crate and the Earth, find the change in the system's total energy, the change in its gravitational potential energy, and the crate's final speed.

Conventions and boundary. Up is positive, the zero of gravitational potential energy sits at the starting height, and the system is crate plus Earth. The rope is outside that boundary, so the rope's work is work done on the system.
By 3.4.C.3, nonzero work on a selected system means energy is transferred between the system and the environment, and by 3.4.B.4 the change in the system's total energy equals that transfer: $\Delta E_{\text{total}} = +120 \, \mathrm{J}$ . Statement 3.4.C.2 does not apply here at all, because its first clause has already failed.
Gravitational term: $\Delta U_g = mg\Delta y = (2.5)(9.8)(+3.0) = +73.5 \, \mathrm{J}$ .
The rest of the transfer has to land in the only other store this system has, by 3.4.B.2: $\Delta K = 120 - 73.5 = 46.5 \, \mathrm{J}$ .
The crate started from rest, so $K_f = 46.5 \, \mathrm{J}$ and $v = \sqrt{2K_f/m} = \sqrt{2(46.5)/2.5} = \sqrt{37.2} = 6.10 \, \mathrm{m/s}$ .
Sense check on what was never needed: the slope angle. The rope's work was given directly, and gravity cares only about the vertical rise, so the geometry of the ramp drops out. Recognizing which given is surplus is a normal part of skill 2.A.

The system's total energy rises by 120 J, its gravitational potential energy by 73.5 J, and the crate reaches $6.1 \, \mathrm{m/s}$ . The rope is the environment doing work on the system, which is 3.4.C.3 with numbers in it: the mechanical energy here is not constant, and it does not need to be, because energy crossed the boundary.

Frequently asked questions

Is mechanical energy always conserved?

No. Mechanical energy, the sum of a system's kinetic and potential energies, is constant only under the two conditions in essential knowledge 3.4.C.2: the work done on the selected system is zero, and there are no nonconservative interactions within the system. Friction inside the boundary or a rope pulling from outside breaks it. Total energy is different: 3.4.C.1 says energy is conserved in all interactions, with no conditions attached.

What is the difference between conservation of energy and conservation of mechanical energy?

Conservation of energy is unconditional. Essential knowledge 3.4.C.1 states that energy is conserved in all interactions. Conservation of mechanical energy is a narrower claim about one chosen system, that its kinetic plus potential energy stays constant, and it holds only when no work is done on the system and nothing nonconservative acts inside it. A block skidding to a stop satisfies the first and violates the second, because its kinetic energy became thermal energy rather than disappearing.

How do you choose the system in an AP Physics 1 energy problem?

Draw the boundary wherever the accounting comes out simplest, then keep it there. Including the Earth turns gravity into an internal interaction with a gravitational potential energy; leaving the Earth out makes gravity an external force doing work on the system. Including the surface a block slides on turns friction into an internal nonconservative interaction instead of external work. Every valid choice gives the same physical answer, and the one rule is never to count the same interaction as work and as potential energy in the same equation.

Where does the energy go when mechanical energy is lost?

Into thermal energy or sound. That is the Topic 3.4 boundary statement in the AP Physics 1 course description: students are expected to know that mechanical energy can be dissipated as thermal energy or sound by nonconservative forces, of which friction and air resistance are the named examples. The energy is not destroyed. It leaves the kinetic-plus-potential ledger, which is exactly how total energy stays conserved while mechanical energy falls.

Can a single object have potential energy?

No. Essential knowledge 3.4.A.1 states that a system composed of only a single object can only have kinetic energy. Potential energy is stored in an interaction, so it needs at least two objects inside the system, which is why the accurate phrasing is that the ball-and-Earth system has gravitational potential energy rather than the ball. Statement 3.4.A.2 adds a second route in: a system that can change its shape reversibly, such as one containing a spring, can also hold potential energy.

Is the conservation of energy equation on the AP Physics 1 formula sheet?

No. The sheet prints the pieces and not the balance: kinetic energy, work as the force component along the displacement times that displacement, the work-energy theorem, spring potential energy, the general gravitational potential energy of two masses, and the change in gravitational potential energy near a surface as the change in kinetic energy equals the sum of the works. You assemble the conservation statement yourself, which is part of why Topic 3.4 lists skill 2.A, deriving a symbolic expression by following a logical mathematical pathway.

What does AP Physics 1 mean by a closed system?

The AP Physics 1 course description does not use that term. Where a textbook says closed or isolated, the CED states the same idea as a choice: 3.4.B.3 says a system may be selected so that the total energy of that system is constant, and 3.4.B.4 says that if the total energy does change, the change equals the energy transferred into or out of the system. So read closed system as a boundary drawn so that no energy crosses it.