AP exam review verified for 2027

AP Physics C: Mechanics Unit 3 Review: Work, Energy, and Power

Q: What topics are covered in AP Physics Mech Unit 3?

AP Physics C: Mechanics Unit 3 covers 5 topics: Translational Kinetic Energy (3.1), Work (3.2), Potential Energy (3.3), Conservation of Energy (3.4), and Power (3.5). The unit builds around the work-energy theorem and conservation principles, which show up repeatedly in later units and on the AP exam. See the full topic breakdown at AP Physics C: Mechanics Unit 3 .

Q: How much of the AP Physics Mech exam is Unit 3?

Unit 3: Work, Energy, and Power makes up 15-25% of the AP Physics C: Mechanics exam, making it one of the heavier-weighted units. That percentage covers Translational Kinetic Energy, Work, Potential Energy, Conservation of Energy, and Power. Expect both multiple-choice and free-response questions that test these concepts.

Q: What's on the AP Physics Mech Unit 3 progress check (MCQ and FRQ)?

The AP Classroom Unit 3 progress check includes both MCQ and FRQ parts drawn from all five Unit 3 topics: Translational Kinetic Energy, Work, Potential Energy, Conservation of Energy, and Power. The MCQ section tests conceptual reasoning and calculation, while the FRQ part asks you to apply the work-energy theorem and conservation of energy to multi-part scenarios. Practice with matched questions at AP Physics C: Mechanics Unit 3 .

Q: How do I practice AP Physics Mech Unit 3 FRQs?

Unit 3 FRQs most often come from Conservation of Energy (3.4) and Work (3.2), asking you to set up energy equations, justify sign conventions, and interpret graphs of force vs. displacement. Practice by writing out full solutions that include a clear energy diagram, labeled variables, and a written justification, not just a numerical answer. Find Unit 3 FRQ practice at AP Physics C: Mechanics Unit 3 .

Q: Where can I find AP Physics Mech Unit 3 practice questions?

The best place to find AP Physics C: Mechanics Unit 3 practice questions, including multiple-choice and practice test sets, is AP Physics C: Mechanics Unit 3 . That page organizes MCQ and FRQ practice by topic, covering Translational Kinetic Energy, Work, Potential Energy, Conservation of Energy, and Power so you can target whichever topic needs the most work.

Q: How should I study AP Physics Mech Unit 3?

Start with Work (3.2) and the work-energy theorem before moving to Potential Energy (3.3) and Conservation of Energy (3.4), since those topics build directly on each other. Sketch energy diagrams for every problem, practice setting up integrals for variable forces, and check your sign conventions carefully. Finish by connecting Power (3.5) to the rate of energy transfer in real scenarios. Use the topic-by-topic resources at AP Physics C: Mechanics Unit 3 to track your progress.

Review AP Physics C: Mechanics Unit 3 to build fluency with work, energy, and power, the tools that let you analyze motion without tracking every force. This unit covers translational kinetic energy, the work integral, potential energy functions, conservation of mechanical energy, and power calculations.

Use the topic guides, FRQ practice, and score calculator on this page to work through every concept before exam day.

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AP exam review verified for 2027

What is AP Physics C: Mechanics unit 3?

What is AP Physics C: Mechanics Unit 3?

Unit 3 builds the energy toolkit: kinetic energy, work as a line integral, potential energy stored by conservative forces, conservation of mechanical energy, and power as the rate of energy transfer. These ideas give you a second method for solving dynamics problems alongside Newton's second law from Unit 2.

Work is the bridge between force and energy

Work quantifies how much energy a force transfers to or from a system. For a constant force, W = Fd cos(theta). For a variable force, you must evaluate the line integral W = integral of F(r) dot dr. The sign of work tells you whether energy enters or leaves the system.

Potential energy lives in the system, not the object

Potential energy is stored in a system of objects interacting through conservative forces such as gravity or a spring. The relationship F_x = -dU/dx lets you extract force information from a U(x) graph, and equilibrium points appear where the slope of U(x) is zero.

Conservation of energy is a bookkeeping tool

When only conservative forces do work, total mechanical energy E = K + U stays constant. When nonconservative forces like friction are present, the work they do equals the change in total mechanical energy. Choosing your system boundary carefully determines which energy transfers appear in your equation.

Why the energy framework matters across the course

The work-energy theorem and conservation of energy reappear in Unit 4 (linear momentum), Unit 5 and 6 (rotational kinetic energy and moment of inertia), and Unit 7 (oscillations and spring potential energy). Understanding the calculus-based definitions here, especially the line integral for work and the derivative relationship between force and potential energy, is essential for every subsequent unit.

AP Physics C: Mechanics unit 3 topics

3.1

Translational Kinetic Energy

Define K = (1/2)mv^2, recognize it as a scalar, and explain why its value depends on the observer's reference frame.

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3.2

Work

Calculate work using W = Fd cos(theta) for constant forces and W = integral F dot dr for variable forces. Distinguish conservative from nonconservative forces and apply the work-energy theorem.

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3.3

Potential Energy

Derive potential energy functions from conservative forces using delta U = -integral F dot dr, extract force from F_x = -dU/dx, and identify equilibrium from U(x) graphs.

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3.4

Conservation of Energy

Apply E = K + U for conservative systems and W_nc = delta E when nonconservative forces act. Choose system boundaries to set up energy equations correctly.

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3.5

Power

Calculate average power with P_avg = W / delta t and instantaneous power with P_inst = dW/dt or P = Fv cos(theta). Convert between watts and other energy-rate units.

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practice snapshot

Hardest AP Physics C: Mechanics unit 3 topics

This snapshot uses Fiveable practice activity to show where students tend to miss questions and which review moves are worth prioritizing first.

61%average MCQ accuracy

Across 2.9k multiple-choice practice attempts for this unit.

2.9kMCQ attempts

Practice activity included in this snapshot.

37%average FRQ score

Across 6 scored free-response attempts for this unit.

Hardest topics in unit 3

MCQ miss rate

3.3

Potential Energy

Review Potential Energy with attention to how the concept appears in AP-style source and evidence questions.

46%747 tries

3.5

Power

Review Power with attention to how the concept appears in AP-style source and evidence questions.

44%349 tries

3.2

Work

Review Work with attention to how the concept appears in AP-style source and evidence questions.

35%567 tries

3.1

Translational Kinetic Energy

Review Translational Kinetic Energy with attention to how the concept appears in AP-style source and evidence questions.

32%689 tries

Unit 3 review notes

3.1

Translational Kinetic Energy

Translational kinetic energy is the energy an object has due to its translational motion. It is always a scalar and always non-negative. Because kinetic energy depends on v^2, doubling speed quadruples K. The value of K depends on the observer's reference frame, so two inertial observers can legitimately measure different values for the same object.

K = (1/2)mv^2: Kinetic energy in joules; m in kg, v in m/s. Scalar quantity, always greater than or equal to zero.
Scalar quantity: K has magnitude only, no direction. You add kinetic energies algebraically, not as vectors.
Frame dependence: Velocity is measured relative to an observer, so K changes with reference frame. There is no single 'correct' value of K for an object.
K vs v graph: A plot of K versus v is a parabola (quadratic). A plot of K versus m is linear at constant v.

A 2 kg object moves at 3 m/s in one frame and 5 m/s in another. Calculate K in each frame and confirm that both values are physically valid.

Quantity	Depends on mass?	Depends on speed?	Scalar or vector?
Kinetic energy K	Yes, linearly	Yes, quadratically	Scalar
Momentum p	Yes, linearly	Yes, linearly	Vector

3.2

Work

Work is the energy transferred into or out of a system by a force acting over a displacement. For a constant force, W = Fd cos(theta), where theta is the angle between force and displacement. For a variable force, W = integral from a to b of F(r) dot dr. The area under a graph of the force component parallel to displacement versus displacement equals the work done. Conservative forces do path-independent work; nonconservative forces like kinetic friction do path-dependent work.

W = integral F dot dr: General definition of work as a line integral. Reduces to Fd cos(theta) when the parallel force component is constant.
Conservative force: A force whose work depends only on initial and final positions, not the path taken. Examples: gravity, spring force. Work around a closed path is zero.
Nonconservative force: A force whose work is path-dependent. Kinetic friction and air resistance are the most common examples; they typically remove mechanical energy from the system.
Work-energy theorem: The net work done on an object equals its change in kinetic energy: W_net = delta K. This holds regardless of whether forces are conservative or not.
Sign of work: Positive work adds energy to the system; negative work removes it. Work is zero when force is perpendicular to displacement.

A spring with k = 200 N/m is compressed 0.10 m from equilibrium. Use the line integral to find the work done by the spring as it returns to equilibrium.

Force type	Path dependent?	Associated potential energy?	Example
Conservative	No	Yes	Gravity, spring
Nonconservative	Yes	No	Kinetic friction, air resistance

3.3

Potential Energy

Potential energy is stored energy associated with the configuration of a system whose objects interact through conservative forces. It is a scalar tied to position. The zero reference point is chosen by the analyst to simplify calculation; only changes in potential energy are physically meaningful. The key calculus relationship is delta U = -integral of F_cf dot dr, or equivalently F_x = -dU/dx in one dimension. A U(x) graph encodes both the potential energy and the force at every position.

delta U = -integral F_cf dot dr: Change in potential energy equals the negative of the work done by the conservative force. This is the defining relationship between conservative forces and potential energy.
F_x = -dU/dx: The conservative force in the x-direction equals the negative slope of the U(x) graph. Force points in the direction of decreasing potential energy.
Gravitational PE (near surface): U_g = mgy, where y is measured from the chosen zero reference. Valid when g is approximately constant.
Elastic PE: U_s = (1/2)k(delta x)^2 for a spring displaced delta x from its natural length.
Equilibrium from U(x): Equilibrium occurs where dU/dx = 0. A local minimum is stable equilibrium; a local maximum is unstable equilibrium.

Given U(x) = 3x^2 - 12x (in joules, x in meters), find the equilibrium position and determine whether it is stable or unstable.

System	Potential energy formula	Zero reference convention
Object near Earth's surface	U_g = mgy	Chosen height y = 0
Spring-mass system	U_s = (1/2)kx^2	Natural length x = 0
Two point masses (gravitation)	U = -Gm1m2/r	Zero at r = infinity

3.4

Conservation of Energy

Mechanical energy is E = K + U. When only conservative forces act within a chosen system and no work is done on the system from outside, E is constant: K_i + U_i = K_f + U_f. When nonconservative forces act, the work they do equals the change in total mechanical energy: W_nc = delta E = delta K + delta U. Choosing the system boundary carefully determines which forces are internal and which do work across the boundary.

E = K + U: Total mechanical energy is the sum of kinetic and potential energies. Conserved only when all internal forces are conservative and no external work is done.
W_nc = delta E: Work done by nonconservative forces equals the change in total mechanical energy. Friction typically makes delta E negative.
System selection: The choice of system boundary determines which energy transfers appear as work done on the system versus internal energy changes.
Energy dissipation: Nonconservative forces convert mechanical energy into thermal energy or sound. This energy is not lost from the universe but is no longer available as mechanical energy.

A 1 kg block slides down a 2 m ramp inclined at 30 degrees with a coefficient of kinetic friction of 0.2. Use energy methods to find the speed at the bottom.

Scenario	Mechanical energy conserved?	Equation to use
Only conservative forces	Yes	K_i + U_i = K_f + U_f
Friction or drag present	No	K_i + U_i + W_nc = K_f + U_f
External force does work	No	W_ext = delta K + delta U

3.5

Power

Power is the rate at which energy is transferred or converted. Average power is P_avg = delta E / delta t or equivalently W / delta t. Instantaneous power is P_inst = dW/dt. When a constant force acts on an object moving at velocity v, instantaneous power simplifies to P_inst = F_parallel * v = Fv cos(theta). The SI unit of power is the watt (1 W = 1 J/s).

P_avg = W / delta t: Average power equals total work divided by elapsed time. Units are watts (W) or joules per second.
P_inst = dW/dt: Instantaneous power is the time derivative of work. Use this when force or velocity varies with time.
P_inst = Fv cos(theta): For a constant force, instantaneous power equals the dot product of force and velocity. Only the component of force parallel to velocity contributes.
Watt: SI unit of power: 1 W = 1 J/s. Kilowatts (kW) are common in applied problems.

A motor exerts a constant 500 N force on a cart moving at 4 m/s at an angle of 30 degrees between force and velocity. Calculate the instantaneous power delivered.

Formula	When to use	Variables needed
P_avg = W / delta t	Total work over a time interval is known	W, delta t
P_avg = delta E / delta t	Energy change over a time interval is known	delta E, delta t
P_inst = dW/dt	Work is a function of time	W(t) expression
P_inst = Fv cos(theta)	Constant force, known velocity	F, v, theta

Practice AP Physics C: Mechanics unit 3 questions

Try AP-style multiple-choice questions and written prompts after you review the notes.

Example AP-style MCQs

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MCQ

AP-style practice question

Question

An object falls through the air subject to a drag force $F_{drag} = -bv$ , where $b = 0.50$ kg/s. The object reaches a constant terminal speed of 20 m/s. What is the rate at which energy is dissipated by the drag force at this speed?

10 W, representing the magnitude of the drag force at the terminal speed.

100 W, representing the energy dissipated using half the velocity squared.

200 W, representing the rate of energy dissipation by the drag force.

400 W, representing the dissipation rate calculated with velocity squared.

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MCQ

AP-style practice question

Question

A satellite moves in an elliptical orbit around Earth from perigee (closest point) to apogee (farthest point). System I is defined as the satellite only. System II is defined as the satellite and Earth. Which of the following correctly describes the change in total mechanical energy for each system during this motion?

System I decreases; System II remains constant

System I remains constant; System II decreases

System I decreases; System II decreases

System I remains constant; System II remains constant

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Example FRQs

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FRQ

Position-dependent force work and energy

1. A block of mass $m$ slides along a frictionless horizontal surface in the +x-direction with initial speed $v_0$ at position $x = 0$ , as shown in Figure 1. A horizontal force $F(x)$ acts on the block in the +x-direction as the block moves. The magnitude of the force varies with position according to $F(x) = F_0 \cos\left(\frac{\pi x}{2L}\right)$ , where $F_0$ is the maximum force and $L$ is a characteristic distance, as shown in Figure 2. The block moves from $x = 0$ to $x = L$ , where the force becomes zero.

Figure 1. Block on a frictionless horizontal surface at x = 0 moving in the +x direction with initial speed v₀ while a position-dependent horizontal force acts to the right; the point x = L is marked to the right where the motion segment ends.

A clean, single-panel physics setup diagram (no graph paper, no grid, no background scenery).

Surface and axis:
- Draw a straight, perfectly horizontal surface line spanning the full width of the panel.
- Above the surface, draw a standard Cartesian x-axis as a thin horizontal axis line coincident with the surface line.
- At the left-of-center portion of the surface, place the origin tick and label it with the visible text "x = 0" directly below the axis line.
- Add an arrowhead on the right end of the axis to indicate the positive direction.
- Place the visible text "+x" just above the arrowhead at the positive end of the axis.

Block:
- Draw a rectangular block resting on the surface line (its bottom edge touching the surface).
- Position the block so that its vertical centerline is exactly aligned above the origin mark labeled "x = 0" (the block is at x = 0).
- Make the block width roughly twice its height.
- Center the visible label "m" inside the block to indicate its mass.

Velocity indication:
- From the center of the block, draw a rightward arrow (horizontal) representing velocity.
- The velocity arrow length should be clearly longer than the block height (so it reads as a vector), and it must point exactly parallel to the surface.
- Label this arrow with the visible text "v₀" placed just above the arrow, centered along the arrow shaft.

Force indication:
- From the right face of the block (starting at the midpoint of the right vertical edge), draw a second rightward horizontal arrow representing the applied force.
- Make the force arrow slightly shorter than the velocity arrow but clearly visible and distinct.
- Label this force arrow with the visible text "F(x)" placed just above the arrow, centered along its shaft.

Marking x = L:
- To the right of the block, along the same axis line, place a single vertical tick mark to indicate the location x = L.
- The x = L tick mark must be separated from the block by a clear blank horizontal distance (at least one full block width of empty space between the block’s right face and the tick mark).
- Label this tick with the visible text "x = L" directly below the axis line.

Clarity constraints:
- No other forces (no normal force, no weight) are drawn.
- No friction arrows are drawn.
- All arrows are solid black with filled arrowheads.
- The only visible text in the diagram is: "x = 0", "+x", "x = L", "m", "v₀", and "F(x)".

Figure 2. Force as a function of position: F(x) = F₀ cos(πx/(2L)), starting at F₀ when x = 0 and decreasing smoothly to 0 at x = L.

A single 2D Cartesian plot of force versus position (no title). Use solid black axes and a solid black curve.

Axes (REQUIRED, exact):
- Horizontal axis label: visible text "Position x (m)" centered beneath the axis.
- Horizontal axis numeric range: from 0 at the origin to L at the right boundary of the plotted region.
- Horizontal axis tick marks: exactly three labeled ticks—"0" at the origin, "L/2" at the midpoint of the axis length, and "L" at the rightmost end of the axis.
- Vertical axis label: visible text "Force F (N)" centered along the vertical axis.
- Vertical axis numeric range: from 0 at the origin up to F₀ at the top boundary of the plotted region.
- Vertical axis tick marks: exactly three labeled ticks—"0" at the origin, "F₀/2" halfway up the axis, and "F₀" at the top of the axis.
- Origin: the intersection of axes is labeled with the visible text "0" (at the lower-left corner of the plotted region).
- Arrowheads: include arrowheads only at the positive ends of both axes (right end of x-axis, top end of y-axis).

Curve (REQUIRED: exact qualitative geometry + required constraints):
- Plot a single smooth curve representing F(x) over the domain from x = 0 to x = L.
- The curve starts at the left boundary on the vertical axis at the maximum force value, exactly at the tick labeled "F₀" when x is at the tick labeled "0".
- At the starting point, the curve has a horizontal tangent (flat, zero slope) as it leaves x = 0.
- From x = 0 moving rightward, the curve decreases monotonically (never increases) and remains entirely above the x-axis until it reaches x = L.
- The curve is concave down (∩-shaped) over the entire interval from x = 0 to x = L, with no inflection points.
- At the midpoint tick labeled "L/2", the curve passes exactly through the mid-height tick labeled "F₀/2" (so the curve goes through the intersection of x = L/2 and F = F₀/2).
- The curve ends exactly on the x-axis at the rightmost tick labeled "L", where the force value is exactly zero.
- At the endpoint at x = L, the curve meets the x-axis with a vertical tangent (the curve is steep there), emphasizing that it approaches the axis sharply at the end.
- Mark the endpoint at x = L with a closed (filled) dot to indicate the value is included and exactly zero there.

Equation annotation:
- Inside the plotting area, place the visible text label "F(x) = F₀ cos(πx/(2L))" near the upper-left region, positioned so it does not overlap the curve.

Styling constraints:
- No grid lines.
- No additional curves.
- Medium line thickness for the curve, slightly thinner for axes.
- Only the specified tick labels appear: 0, L/2, L on the x-axis and 0, F₀/2, F₀ on the y-axis.

Derive an expression for the speed $v_f$ of the block at position $x = L$ . Express your answer in terms of $m$ , $v_0$ , $F_0$ , $L$ , and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

ii.

During the time interval when the block moves from $x = 0$ to $x = L$ , the force $F(x)$ does work on the block. Derive an expression for the work $W$ done by the force on the block during this displacement. Express your answer in terms of $F_0$ , $L$ , and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

Calculate the numerical value of the average power delivered to the block by the force $F(x)$ as the block moves from $x = 0$ to $x = L$ . The block has mass $m = 0.50$ kg, initial speed $v_0 = 3.0$ m/s, the maximum force is $F_0 = 8.0$ N, and the distance is $L = 2.0$ m. The time taken for the block to travel from $x = 0$ to $x = L$ is $\Delta t = 0.40$ s.

Describe how the selection of the block as the system, rather than the block-Earth system, affects whether the mechanical energy of the system is conserved as the block moves from $x = 0$ to $x = L$ . Justify your answer. The horizontal surface is frictionless and the block moves in a horizontal plane at constant height above the ground.

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FRQ

Variable force, work-energy theorem, mass comparison

4. A block of mass m = 2.0 kg is initially at rest on a horizontal frictionless surface. A time-varying horizontal force F is applied to the block. The force varies with position x according to the equation F(x) = F₀(1 - x/d), where F₀ = 12 N and d = 3.0 m, as shown in Figure 1. The force is applied in the positive x-direction as the block moves from x = 0 to x = d.

Figure 1: Applied horizontal force F as a function of position x for F(x) = F₀(1 − x/d), with F₀ = 12 N and d = 3.0 m

A single, clean 2D Cartesian graph showing force as a function of position.

Axes and formatting (must be exact):
- Horizontal axis label: "Position x (m)".
- Horizontal axis numerical range: from 0 to 3.0.
- Horizontal axis tick marks: every 0.5 m, labeled 0, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0.
- Vertical axis label: "Force F (N)".
- Vertical axis numerical range: from 0 to 12.
- Vertical axis tick marks: every 2 N, labeled 0, 2, 4, 6, 8, 10, 12.
- The origin is explicitly labeled "0" at the intersection of the axes.
- Arrowheads appear only at the positive ends of both axes.
- No gridlines.
- No title text inside the plotting area (caption is outside the graph).

Curve/line (must be exact and unambiguous):
- Plot a single solid black line (medium thickness) representing F versus x.
- The plotted relation is a perfectly straight line segment (no curvature).
- The line starts on the y-axis at the top tick labeled 12 N when x is at the left boundary value 0 m.
- The line decreases with constant slope as x increases (uniform linear decrease).
- The line ends on the x-axis exactly at the right boundary value 3.0 m where the force is exactly 0 N.
- Use closed (filled) endpoints at both ends of the line segment to indicate the endpoints are included.

Key mathematical constraints made visually explicit:
- At x = 0 m, the force equals the maximum shown value 12 N.
- At x = 3.0 m, the force equals 0 N (the line touches the x-axis only at this rightmost endpoint).
- The line crosses intermediate tick intersections consistently with a linear rule, such that at the midpoint of the x-range (x = 1.5 m) the force is exactly halfway between 12 N and 0 N, i.e., aligned with the y tick labeled 6 N.

Only visible text in the plot area should be axis labels and tick numbers; no equation text is printed on the graph.

The 2.0 kg block reaches a final speed v₁ at position x = d. The 4.0 kg block reaches a final speed v₂ at position x = d.

Indicate whether v₁ is greater than, less than, or equal to v₂ by writing one of the following.

v₁ > v₂
v₁ < v₂
v₁ = v₂

Justify your answer using qualitative reasoning beyond referencing equations.

Derive an expression for the speed v of the 2.0 kg block when it reaches position x = d. Express your answer in terms of m, F₀, d, and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information. The 2.0 kg block moves from x = 0 to x = d under the influence of the position-dependent force.

Indicate whether P₁ is greater than, less than, or equal to P₂. The average power delivered to the 2.0 kg block as it moves from x = 0 to x = 1.5 m is P₁. The average power delivered to the 2.0 kg block as it moves from x = 1.5 m to x = 3.0 m is P₂.

Briefly justify your answer.

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FRQ

Variable force work and kinetic energy changes

2. A block of mass m = 3.0 kg starts from rest at position x = 0 m on a frictionless horizontal surface. A variable horizontal force F is applied to the block, causing it to move in the positive x-direction. The magnitude of the applied force as a function of position is given by F(x) = F₀(1 - x/d), where F₀ = 24 N and d = 4.0 m, as shown in Figure 1. The force decreases linearly from F₀ at x = 0 to zero at x = d.

Figure 1: Applied horizontal force magnitude F as a function of position x for F(x) = F₀(1 − x/d), with F₀ = 24 N and d = 4.0 m

Single-panel Cartesian graph.

Axes:
- Horizontal axis label: "x (m)".
- Horizontal axis range: from 0 to 4.0.
- Horizontal tick marks: every 0.5 m, labeled at 0, 1.0, 2.0, 3.0, and 4.0 (minor unlabeled ticks at 0.5, 1.5, 2.5, 3.5).
- Vertical axis label: "F (N)".
- Vertical axis range: from 0 to 24.
- Vertical tick marks: every 4 N, labeled 0, 4, 8, 12, 16, 20, 24.
- Origin: the axes intersect at the lower-left corner and the number "0" is printed at the intersection.
- Arrows: arrowhead on the positive end of the x-axis (to the right) and on the positive end of the F-axis (upward).

Curve shape and behavior:
- A single solid black straight line segment represents the force function.
- The line begins on the vertical axis at the very top tick mark labeled 24 N (this is at x = 0).
- The line decreases with constant slope (perfectly linear, no curvature) as x increases.
- The line crosses the horizontal axis exactly at the rightmost x tick labeled 4.0 m, where the force equals 0 N.
- The line has no points beyond x = 4.0 m (it stops exactly at the boundary at x = 4.0).

Styling:
- Line: solid black, medium thickness.
- No gridlines.
- No legend.
- No additional equation text printed on the plot other than axis labels and tick labels.

Figure 2: Energy bar chart template for Scenario 2 showing work by the applied force W_F, kinetic energy K, and work by the resistive force W_f at x = 2.0 m and x = 4.0 m

Two-group vertical bar chart template with a shared energy axis.

Axes:
- Vertical axis label: "Energy (J)".
- Vertical axis range: from −30 J to +60 J.
- Vertical tick marks: every 10 J, labeled −30, −20, −10, 0, 10, 20, 30, 40, 50, 60.
- A bold horizontal zero-energy line is drawn at the tick labeled 0.
- Horizontal axis label: "Quantity".
- Horizontal axis shows SIX category positions (tick marks at the center of each category).
- Categories (left to right, printed under each bar):
1) "W_F at x = 2.0 m"
2) "K at x = 2.0 m"
3) "W_f at x = 2.0 m"
4) "W_F at x = 4.0 m"
5) "K at x = 4.0 m"
6) "W_f at x = 4.0 m"
- Origin labeling: the y-axis has the tick label "0" at the zero line; there is no (0,0) point label because the x-axis is categorical.
- Arrows: no arrows on axes (standard bar-chart style).

Bar specifications (all bars are present as light-gray placeholders to be shaded by students):
- Each of the six bars has identical width (uniform width), with small equal gaps between adjacent bars.
- Each bar has a black outline, medium stroke.
- Placeholder fill: white (unshaded) interior so students can shade.

Required numerical bar heights for the correct completed chart (these values determine the proportional shading targets):
- For x = 2.0 m:
- W_F bar extends upward from 0 to +18 J.
- K bar extends upward from 0 to +6 J.
- W_f bar extends downward from 0 to −12 J (negative work).
- For x = 4.0 m:
- W_F bar extends upward from 0 to +48 J.
- K bar extends upward from 0 to +24 J.
- W_f bar extends downward from 0 to −24 J.

Zero-energy representation rule:
- Any quantity equal to 0 would be shown as a thin horizontal line segment exactly on the 0 J line in that category position (include this instruction as part of the diagram even though none of the six target values is zero).

Error bars (added to satisfy the provided bar-graph requirements):
- Each bar includes a vertical error bar with a central marker at the bar’s mean height and capped ends.
- Error-bar cap width: one-third of the bar width.
- Exact error-bar endpoints:
- W_F at x = 2.0 m (mean +18 J): error bar from +16 J to +20 J.
- K at x = 2.0 m (mean +6 J): error bar from +5 J to +7 J.
- W_f at x = 2.0 m (mean −12 J): error bar from −13 J to −11 J.
- W_F at x = 4.0 m (mean +48 J): error bar from +44 J to +52 J.
- K at x = 4.0 m (mean +24 J): error bar from +22 J to +26 J.
- W_f at x = 4.0 m (mean −24 J): error bar from −26 J to −22 J.

Styling:
- No gridlines.
- No title beyond the caption.
- Text uses consistent physics notation with subscripts: W_F and W_f.

An energy bar chart can be used to represent the work W_F done by the applied force, the kinetic energy K of the block, and the work W_f done by the resistive force (represented as a negative quantity). On the energy bar chart in Figure 2, draw two sets of shaded bars: In Scenario 2, the block moves from x = 0 to x = d under both the variable force F(x) and a constant resistive force f = 6.0 N.

One set representing the energies when the block is at position x = 2.0 m
A second set representing the energies when the block is at position x = 4.0 m

The height of the shaded bars should be proportional to the magnitudes of W_F, K, and |W_f| at each position. Any energy equal to zero should be represented by a distinct line on the zero-energy line.

Derive an expression for the speed v of the block when it reaches position x = 2.0 m. Express your answer in terms of m, F₀, d, and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information. In Scenario 1, the block moves from x = 0 to x = d with no friction.

Figure 3: Velocity v versus position x for Scenario 1 (frictionless, variable applied force)

Single-panel Cartesian graph showing a specific concave-down velocity curve consistent with the work–energy result for F(x) = 24(1 − x/4.0) acting on a 3.0 kg block starting from rest.

Axes:
- Horizontal axis label: "x (m)".
- Horizontal axis range: from 0 to 4.0.
- Horizontal tick marks: every 0.5 m, labeled at 0, 1.0, 2.0, 3.0, 4.0.
- Vertical axis label: "v (m/s)".
- Vertical axis range: from 0 to 6.0.
- Vertical tick marks: every 0.5 m/s, labeled at 0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0.
- Origin: labeled "0" at the axes intersection.
- Arrows: arrowhead on the positive end of the x-axis and on the positive end of the v-axis.

Curve shape and behavior:
- A single smooth solid black curve.
- Starts exactly at v = 0 on the left boundary at x = 0 with a closed dot at the origin.
- Immediately increases steeply for small x, then the slope steadily decreases as x increases (concave down everywhere on the interval).
- The curve passes exactly through the tick intersection corresponding to x = 1.0 m and v = 3.0 m/s.
- The curve passes exactly through the tick intersection corresponding to x = 2.0 m and v = 4.0 m/s.
- The curve passes exactly through the tick intersection corresponding to x = 3.0 m and v = 5.0 m/s.
- Ends at the right boundary at x = 4.0 m with v = 5.66 m/s shown by placing the endpoint slightly above the 5.5 tick and below the 6.0 tick; label the y-value "5.66" next to the endpoint tick level (as a small annotation) while keeping the axis ticks unchanged.
- No maxima or minima within the interior of the graph: v increases monotonically from left to right.
- No inflection points: concave down for the entire curve.

Styling:
- Curve: solid black, medium thickness.
- No gridlines.
- No legend.
- No equation text inside the plot.

Figure 4: Axes for instantaneous power P delivered by the applied force versus position x (Scenario 1, frictionless)

Single-panel Cartesian axes intended for a student sketch of power versus position.

Axes:
- Horizontal axis label: "x (m)".
- Horizontal axis range: from 0 to 4.0.
- Horizontal tick marks: every 0.5 m, labeled at 0, 1.0, 2.0, 3.0, 4.0.
- Vertical axis label: "P (W)".
- Vertical axis range: from 0 to 80.
- Vertical tick marks: every 10 W, labeled 0, 10, 20, 30, 40, 50, 60, 70, 80.
- Origin: labeled "0" at the axes intersection.
- Arrows: arrowhead on the positive end of the x-axis and on the positive end of the P-axis.

Reference markers to enforce key values (light, non-data guide marks):
- A faint open circle marker on the origin to indicate the power starts at zero at x = 0.
- A faint guide tick mark (not a full gridline) at x = 4.0 on the x-axis emphasizing the endpoint.

Intended curve constraints for the student sketch (printed as non-numeric visual guides only, not as equations):
- The sketched curve must start at P = 0 at x = 0.
- The sketched curve must return to P = 0 at x = 4.0.
- The curve must have a single interior maximum (one peak) between x = 0 and x = 4.0.

Styling:
- Axes are black.
- No gridlines.
- No pre-drawn power curve (blank axes only, aside from the faint origin marker and faint endpoint emphasis).

On the axes in Figure 4, sketch a graph of the instantaneous power P delivered by the applied force as a function of position x as the block moves from x = 0 to x = 4.0 m in Scenario 1. Your sketch should show: In Scenario 1, the velocity of the block as a function of position is shown in Figure 3. The instantaneous power P delivered by the applied force can be calculated as P = Fv.

The correct initial behavior at x = 0
The location of any maximum or minimum values
The correct final value at x = 4.0 m

Describe one specific feature of the power vs. position graph in Scenario 2 that would differ from the graph you sketched for Scenario 1 in Figure 4. In your description, explicitly state whether the feature increases, decreases, shifts, or changes in another specific way. A student analyzes both Scenario 1 and Scenario 2 and creates graphs of instantaneous power P delivered by the applied force as a function of position x for each scenario.

Justify your answer using physics principles related to work, energy, and power.

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Key terms

Term	Definition
scalar	A physical quantity described by magnitude only, without direction. Both kinetic energy and work are scalars.

Common unit 3 mistakes

Treating work as a vector

Work is a scalar. The dot product F dot dr produces a signed number, not a vector. Students sometimes try to add work values as vectors, which gives wrong answers.

Forgetting the negative sign in F_x = -dU/dx

The conservative force equals the negative derivative of potential energy. Dropping the negative sign reverses the direction of the force and leads to incorrect equilibrium analysis.

Using conservation of mechanical energy when friction is present

K_i + U_i = K_f + U_f only holds when all forces doing work are conservative. If kinetic friction or drag acts, you must include W_nc = delta E or the answer will be too large.

Confusing average power with instantaneous power

P_avg = W / delta t gives the average rate over an interval. P_inst = Fv cos(theta) gives the rate at one instant. Using the average formula when instantaneous power is asked, or vice versa, produces incorrect results.

Setting the wrong zero for potential energy

The zero reference for U is a free choice, but it must stay consistent throughout a single problem. Switching the reference mid-calculation changes delta U and breaks the energy equation.

How this unit shows up on the AP exam

Deriving expressions using calculus

AP Physics C: Mechanics free-response questions frequently ask you to derive a symbolic expression for work, potential energy, or power using integration or differentiation. You should be comfortable setting up the line integral W = integral F dot dr for a given force function and applying F_x = -dU/dx to extract force from a potential energy expression.

Justifying energy conservation claims

Exam questions often ask you to explain whether mechanical energy is conserved in a given scenario and to justify your answer. You need to identify which forces act, classify them as conservative or nonconservative, and state how that classification determines whether E = K + U is constant or whether W_nc = delta E applies.

Interpreting graphs of energy and force

Questions may present a U(x) graph and ask you to identify equilibrium positions, determine the direction of force at a given point, or sketch the corresponding F(x) graph. You may also be asked to find work as the area under an F vs displacement curve or to read instantaneous power from a slope on an energy versus time graph.

Final unit 3 review checklist

Final Unit 3 review checklistUse this list to confirm you can handle every major skill before the exam.
Calculate K and explain frame dependenceApply K = (1/2)mv^2, confirm the scalar nature of kinetic energy, and explain why two inertial observers can measure different values for the same object.
Evaluate work using the line integralSet up and solve W = integral F dot dr for variable forces, including spring and gravitational forces. Find work graphically as the area under an F_parallel vs displacement curve.
Derive and use potential energy functionsUse delta U = -integral F_cf dot dr to derive U(x) for gravity and springs. Apply F_x = -dU/dx to recover force from a given potential energy function.
Identify equilibrium from U(x) graphsLocate points where dU/dx = 0, classify them as stable (local minimum) or unstable (local maximum), and sketch the corresponding force versus position.
Apply conservation of energy with and without frictionWrite K_i + U_i = K_f + U_f for conservative systems and include W_nc = delta E when friction or drag is present. Justify your system selection explicitly.
Calculate average and instantaneous powerUse P_avg = W / delta t and P_inst = Fv cos(theta). Derive an instantaneous power expression using calculus when force or velocity is given as a function of time.

How to study unit 3

Step 1: Kinetic energy and the work-energy theorem (Topics 3.1-3.2)Start by confirming you can calculate K = (1/2)mv^2 and explain its scalar nature and frame dependence. Then practice setting up the work integral for both constant and variable forces, including spring forces. Verify the work-energy theorem by checking that W_net = delta K in several examples. Use the Topic 3.1 and 3.2 guides to review the line integral setup.

Step 2: Potential energy functions and U(x) graphs (Topic 3.3)Practice deriving U(x) from a given conservative force using delta U = -integral F dot dr for gravity and spring systems. Then reverse the process: given a U(x) expression or graph, find F_x = -dU/dx and identify equilibrium points. Sketch force versus position from a U(x) graph. Use the Topic 3.3 guide for worked examples with gravitational and elastic potential energy.

Step 3: Conservation of energy problems (Topic 3.4)Work through problems that require choosing a system, writing the energy equation, and solving for an unknown speed, height, or compression. Practice both the conservative case (K_i + U_i = K_f + U_f) and the nonconservative case (W_nc = delta E). Use the Topic 3.4 guide and FRQ practice to build fluency with multi-step energy problems.

Step 4: Power calculations (Topic 3.5)Practice computing average power from total work and time, and instantaneous power using P = Fv cos(theta). Try deriving a power expression as a function of time when force or velocity varies. Use the Topic 3.5 guide to review the calculus-based definition P_inst = dW/dt.

Step 5: Full-unit integration and exam practiceSolve multi-concept problems that combine work, potential energy, conservation of energy, and power in a single scenario. Use the available FRQ practice to rehearse writing complete justifications. Use the AP score calculator to estimate your current scoring range and identify which topics need more attention.

More ways to review

Topic study guides

Open the individual guides for Unit 3 when you want a closer review of one topic.

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Practice questions

Use AP-style practice after you review the notes so you can check what you understand.

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FRQ practice

Practice free-response reasoning and compare your answer with scoring guidance.

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Cheatsheets

Use unit cheatsheets for a quick visual review after you work through the notes.

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Score calculator

Estimate your broader AP score goal after you review the course and exam format.

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Frequently Asked Questions

What topics are covered in AP Physics Mech Unit 3?

AP Physics C: Mechanics Unit 3 covers 5 topics: Translational Kinetic Energy (3.1), Work (3.2), Potential Energy (3.3), Conservation of Energy (3.4), and Power (3.5). The unit builds around the work-energy theorem and conservation principles, which show up repeatedly in later units and on the AP exam. See the full topic breakdown at AP Physics C: Mechanics Unit 3.

How much of the AP Physics Mech exam is Unit 3?

Unit 3: Work, Energy, and Power makes up 15-25% of the AP Physics C: Mechanics exam, making it one of the heavier-weighted units. That percentage covers Translational Kinetic Energy, Work, Potential Energy, Conservation of Energy, and Power. Expect both multiple-choice and free-response questions that test these concepts.

What's on the AP Physics Mech Unit 3 progress check (MCQ and FRQ)?

The AP Classroom Unit 3 progress check includes both MCQ and FRQ parts drawn from all five Unit 3 topics: Translational Kinetic Energy, Work, Potential Energy, Conservation of Energy, and Power. The MCQ section tests conceptual reasoning and calculation, while the FRQ part asks you to apply the work-energy theorem and conservation of energy to multi-part scenarios. Practice with matched questions at AP Physics C: Mechanics Unit 3.

How do I practice AP Physics Mech Unit 3 FRQs?

Unit 3 FRQs most often come from Conservation of Energy (3.4) and Work (3.2), asking you to set up energy equations, justify sign conventions, and interpret graphs of force vs. displacement. Practice by writing out full solutions that include a clear energy diagram, labeled variables, and a written justification, not just a numerical answer. Find Unit 3 FRQ practice at AP Physics C: Mechanics Unit 3.

Where can I find AP Physics Mech Unit 3 practice questions?

The best place to find AP Physics C: Mechanics Unit 3 practice questions, including multiple-choice and practice test sets, is AP Physics C: Mechanics Unit 3. That page organizes MCQ and FRQ practice by topic, covering Translational Kinetic Energy, Work, Potential Energy, Conservation of Energy, and Power so you can target whichever topic needs the most work.

How should I study AP Physics Mech Unit 3?

Start with Work (3.2) and the work-energy theorem before moving to Potential Energy (3.3) and Conservation of Energy (3.4), since those topics build directly on each other. Sketch energy diagrams for every problem, practice setting up integrals for variable forces, and check your sign conventions carefully. Finish by connecting Power (3.5) to the rate of energy transfer in real scenarios. Use the topic-by-topic resources at AP Physics C: Mechanics Unit 3 to track your progress.

Ready to review Unit 3?Start with the notes, check the topic cards, and use the practice or resource links when they are available for this course.

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