AP Physics 1 Unit 3 Notes: Building Understanding of Work, Energy, and Power

Work as Energy Transfer

What “work” means (and what it does not mean)

In everyday language, “work” can mean effort. In physics, work has a very specific meaning: it is a way that energy is transferred into or out of a system by a force acting through a displacement. That last phrase matters—if there is no displacement, no work is done (in the physics sense), even if you feel tired.

For a constant force, the work done by that force depends on three things:

1. The magnitude of the force.
2. The amount of displacement.
3. The angle between the force and the displacement.

If the force points in the same direction as the displacement, the force transfers energy into the object (positive work). If the force points opposite the displacement, it removes energy (negative work). If the force is perpendicular to the displacement, it transfers no energy (zero work) even though the force might be large.

Work by a constant force

When a constant force acts on an object that moves in a straight line, the work done by that force is

W = Fd\cos\theta

• W is work (joules).
• F is the magnitude of the force (newtons).
• d is displacement magnitude (meters).
• \theta is the angle between the force vector and the displacement vector.

A useful interpretation is that only the component of the force parallel to the displacement does work. The “parallel component” is F\cos\theta, so you can also think:

W = F_{\parallel} d

where F_{\parallel} = F\cos\theta.

Sign of work: The cosine term automatically handles sign.

• If 0^\circ \le \theta < 90^\circ, then \cos\theta > 0 and W > 0.
• If \theta = 90^\circ, then W = 0.
• If 90^\circ < \theta \le 180^\circ, then \cos\theta < 0 and W < 0.

Which forces “count” for work?

Any force can do work if it has a component along the displacement. In AP Physics 1, common forces to analyze include:

• Gravity (weight)
• Normal force
• Tension
• Applied force
• Kinetic friction
• Spring force

A common misconception is “the normal force never does work.” It often does zero work (for example, a block sliding on a level surface, because the normal is perpendicular to displacement), but it can do work if the surface pushes along the direction of motion (for example, a person riding up an accelerating elevator platform).

Work on a system vs. work by a system

It helps to be explicit about what object or system you mean:

• “Work done on the object by a force” is energy transferred to that object.
• “Work done by the object” can mean energy the object transfers to something else.

On free-response questions, you can avoid confusion by writing something like: “Work done by friction on the block” or “Work done by the applied force on the cart.”

Work from graphs: force vs. position

AP Physics 1 frequently uses graphs to represent work conceptually. If you have a graph of force (vertical axis) vs. position (horizontal axis), then the work done by that force over a displacement is the area under the curve.

For a constant force, that area is a rectangle with area F\Delta x (and the sign comes from whether the force is above or below the axis). For a changing force, the area can be found by geometry (triangles, trapezoids) when the graph is piecewise linear.

Worked example 1: angled force pulling a crate

A crate is pulled with a force of magnitude F = 50\ \text{N} at an angle \theta = 30^\circ above the horizontal over a horizontal displacement d = 10\ \text{m}.

The work done by the pulling force is

W = Fd\cos\theta
W = (50)(10)\cos 30^\circ
W = 500\left(\frac{\sqrt{3}}{2}\right)
W \approx 433\ \text{J}

Notice you only used the displacement along the direction of motion (horizontal), so the relevant component of the pulling force is horizontal.

Worked example 2: zero work with a perpendicular force

A satellite moves in a circular orbit at constant speed. Gravity points toward the center, while the satellite’s instantaneous displacement is tangential. The angle between gravity and displacement is 90^\circ, so

W = Fd\cos 90^\circ = 0

This is why (in the idealized circular orbit) gravity changes the direction of velocity but not the speed—it does no work.

Exam Focus

• Typical question patterns:
  • Compute work by an applied force at an angle, often with friction present.
  • Use a F vs. x graph to find work as an area (including sign).
  • Decide which forces do zero work based on direction relative to displacement.
• Common mistakes:
  • Using Fd instead of Fd\cos\theta when the force is angled.
  • Mixing up distance traveled with displacement direction (work uses the displacement direction).
  • Assuming the normal force always does zero work, even in situations where displacement has a component along the normal.

Kinetic Energy and the Work-Energy Theorem

Kinetic energy: energy of motion

Kinetic energy is the energy associated with motion. For an object of mass m moving with speed v,

K = \frac{1}{2}mv^2

Kinetic energy is a scalar (it has no direction). Speed matters a lot because of the square: doubling speed quadruples kinetic energy.

Why kinetic energy matters

Forces can be complicated to track over distance, but energy often gives you a cleaner way to connect “cause” (forces doing work) to “effect” (changes in speed). Instead of tracking acceleration at every moment, you can often compare energies at the beginning and end.

The work-energy theorem

The work-energy theorem links net work to the change in kinetic energy:

W_{\text{net}} = \Delta K

where

\Delta K = K_f - K_i

This is not “conservation of energy” yet—it is a direct consequence of Newton’s second law for motion along a path. It says: if the net work done on an object is positive, its kinetic energy increases (it speeds up); if net work is negative, it slows down.

A key point: W_{\text{net}} means the sum of work by all forces acting on the object.

Connecting to forces: net work vs. work by individual forces

If multiple forces act, then

W_{\text{net}} = \sum W_i

For many AP problems, it’s helpful to compute work by each force (applied, friction, gravity, etc.) and add them.

Worked example 1: net work and speed change

A block of mass m = 2.0\ \text{kg} starts from rest and is pushed with a constant horizontal force F = 10\ \text{N} over d = 3.0\ \text{m} on a frictionless surface.

• Work by the applied force:

W_F = Fd\cos 0^\circ = (10)(3)(1) = 30\ \text{J}

• Normal force and weight do zero work (perpendicular to displacement).

So

W_{\text{net}} = 30\ \text{J}

By the work-energy theorem:

W_{\text{net}} = \Delta K
30 = \frac{1}{2}mv_f^2 - 0
30 = \frac{1}{2}(2.0)v_f^2
30 = v_f^2
v_f = \sqrt{30} \approx 5.48\ \text{m/s}

This example shows a typical AP move: compute work from forces, then convert to a speed using kinetic energy.

Worked example 2: friction and negative work

A block slides across a rough surface with kinetic friction force magnitude f_k = 4\ \text{N} over a displacement d = 5\ \text{m}. Friction points opposite motion.

Work by friction:

W_f = f_k d\cos 180^\circ = (4)(5)(-1) = -20\ \text{J}

The negative sign means friction removes mechanical energy (it transfers energy into thermal energy of the surfaces).

Exam Focus

• Typical question patterns:
  • Use W_{\text{net}} = \Delta K to find a final speed after forces act over a distance.
  • Compute work contributions from multiple forces and interpret the sign of each.
  • Conceptual prompts: “Does the object speed up or slow down?” based on net work.
• Common mistakes:
  • Using only the applied force’s work and forgetting other forces (especially friction).
  • Confusing velocity and speed in kinetic energy (kinetic energy uses v^2).
  • Treating “work done by friction” as automatically positive (it is usually negative when friction opposes motion).

Potential Energy and Conservative Forces

What potential energy represents

Potential energy is energy stored due to the configuration of a system. It is not “energy an object has because it is high” in isolation—rather, it belongs to an interaction (like Earth-object gravity) or to a system (like a spring-object system).

In AP Physics 1, you mainly use:

• Gravitational potential energy near Earth’s surface
• Elastic potential energy in springs

Conservative forces: the key idea

A conservative force is a force for which the work done depends only on the initial and final positions, not on the path taken. When only conservative forces do work, mechanical energy can be conserved.

Two important consequences of conservative forces:

1. The net work done around a closed path is zero.
2. You can define a potential energy function U such that the work done by the conservative force is the negative change in potential energy.

That relationship is

W_c = -\Delta U

where W_c is the work done by the conservative force and \Delta U = U_f - U_i.

Gravitational potential energy near Earth

Near Earth’s surface (uniform gravitational field), the gravitational potential energy of the Earth-object system changes by

\Delta U_g = mg\Delta y

• m is mass.
• g is gravitational field strength (near Earth, approximately 9.8\ \text{m/s}^2).
• \Delta y is change in vertical position (final minus initial).

If an object moves up (positive \Delta y), U_g increases. If it moves down, U_g decreases.

The corresponding work done by gravity is

W_g = -\Delta U_g

So, when an object drops, gravity typically does positive work and potential energy decreases.

Common conceptual trap: Students sometimes think U_g = mgy is “the formula,” but the more fundamental idea is the _change_ \Delta U_g = mg\Delta y. The zero level of potential energy is a choice; the change is what affects physics.

Elastic potential energy (springs)

A spring exerts a restoring force described by Hooke’s law (for ideal springs):

F_s = kx

• k is the spring constant.
• x is the spring’s displacement from equilibrium (stretch or compression magnitude).

Because the force changes with position, you can’t use W = Fd\cos\theta with a single constant F unless you use average force or area under a force-position graph. The energy approach is usually simpler: the elastic potential energy stored in a spring is

U_s = \frac{1}{2}kx^2

The change in elastic potential energy is

\Delta U_s = \frac{1}{2}k\left(x_f^2 - x_i^2\right)

As with gravity, the work done by the spring force is

W_s = -\Delta U_s

Worked example 1: gravitational potential energy change

A 3.0\ \text{kg} backpack is lifted upward by \Delta y = 1.5\ \text{m}.

Change in gravitational potential energy:

\Delta U_g = mg\Delta y = (3.0)(9.8)(1.5) = 44.1\ \text{J}

Gravity’s work is

W_g = -\Delta U_g = -44.1\ \text{J}

That negative work matches the idea that gravity opposes the upward displacement.

Worked example 2: spring energy

A spring with k = 200\ \text{N/m} is compressed from equilibrium by x = 0.10\ \text{m}.

Stored elastic potential energy:

U_s = \frac{1}{2}kx^2 = \frac{1}{2}(200)(0.10)^2 = 1.0\ \text{J}

Even a small compression can store noticeable energy when k is large.

Exam Focus

• Typical question patterns:
  • Use \Delta U_g = mg\Delta y in energy conservation problems (ramps, pendulums, drops).
  • Use U_s = \frac{1}{2}kx^2 to connect spring compression to speed after release.
  • Identify whether a force is conservative (gravity, spring) or nonconservative (friction).
• Common mistakes:
  • Getting the sign wrong for \Delta y (always define up as positive and stick with it).
  • Treating x as “distance traveled” rather than displacement from equilibrium of the spring.
  • Forgetting that potential energy belongs to a system (Earth-object, spring-object), not just the object.

Conservation of Mechanical Energy (and How to Use It)

Mechanical energy: kinetic plus potential

Mechanical energy is the sum of kinetic energy and potential energies that come from conservative interactions:

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

In many AP Physics 1 situations, U might include gravitational potential energy U_g and/or spring potential energy U_s.

When mechanical energy is conserved

Mechanical energy is conserved when:

• The only forces doing work are conservative forces (like gravity and springs), or
• Any nonconservative forces present do zero work (for example, the normal force in many sliding problems)

When mechanical energy is conserved:

K_i + U_i = K_f + U_f

This is powerful because it connects states without needing time, acceleration, or detailed force analysis along the path.

A systematic way to solve energy conservation problems

When you use energy conservation, you are essentially comparing two “snapshots” of the motion.

A consistent process is:

1. Choose your system (often object + Earth, or object + spring).
2. Choose initial and final states.
3. Write expressions for K and the relevant U terms at each state.
4. Set K_i + U_i = K_f + U_f.
5. Solve for the unknown.

A big advantage: you can choose the zero of potential energy wherever it simplifies the math (often at the lowest point).

Example 1: ramp without friction

A block starts from rest at height h = 2.0\ \text{m} above the bottom of a frictionless ramp. Find its speed at the bottom.

Choose the system: block + Earth. No friction, so mechanical energy is conserved.

Initial:

• K_i = 0
• Choose U_{g,i} = mgh

Final at bottom:

• U_{g,f} = 0 (by choice)
• K_f = \frac{1}{2}mv^2

Conservation:

K_i + U_{g,i} = K_f + U_{g,f}
0 + mgh = \frac{1}{2}mv^2 + 0

Mass cancels:

gh = \frac{1}{2}v^2
v^2 = 2gh
v = \sqrt{2gh} = \sqrt{2(9.8)(2.0)} \approx 6.26\ \text{m/s}

This “mass cancels” result is common: in idealized frictionless gravitational motion, speed depends on height change, not mass.

Example 2: pendulum speed at the bottom

A pendulum bob is released from rest from a height difference \Delta y = 0.50\ \text{m} above its lowest point. Neglect air resistance. Find speed at the bottom.

Same structure as the ramp:

mg\Delta y = \frac{1}{2}mv^2

So

v = \sqrt{2g\Delta y} = \sqrt{2(9.8)(0.50)} \approx 3.13\ \text{m/s}

A conceptual point: tension does no work here because it is perpendicular to the arc displacement at each moment.

Example 3: spring launch on a frictionless surface

A block of mass m = 0.50\ \text{kg} is pushed against a spring of constant k = 400\ \text{N/m}, compressing it by x = 0.10\ \text{m}. The block is released on a frictionless horizontal surface. Find the speed when the spring returns to equilibrium.

Choose system: block + spring. No friction, horizontal so gravitational potential energy does not change.

Initial:

• K_i = 0
• U_{s,i} = \frac{1}{2}kx^2

Final at equilibrium:

• U_{s,f} = 0
• K_f = \frac{1}{2}mv^2

Conservation:

\frac{1}{2}kx^2 = \frac{1}{2}mv^2

Solve:

v^2 = \frac{k}{m}x^2
v = x\sqrt{\frac{k}{m}} = (0.10)\sqrt{\frac{400}{0.50}}
v = 0.10\sqrt{800} \approx 2.83\ \text{m/s}

Interpreting energy bar charts (a high-value AP skill)

AP Physics 1 often asks you to reason qualitatively with energy bar charts or similar representations. The goal is to track where energy is stored and how it transfers.

A helpful mindset:

• Bars for K, U_g, U_s represent storage.
• A bar for thermal energy (often written as E_{\text{th}}) represents energy dissipated due to friction.
• “Work by external forces” represents energy transferred into or out of the system.

A frequent misconception is that friction “destroys” energy. In physics, energy is conserved, but friction converts mechanical energy into thermal energy.

Exam Focus

• Typical question patterns:
  • Use K_i + U_i = K_f + U_f to find speeds or heights in frictionless motion.
  • Decide whether mechanical energy is conserved by identifying which forces do work.
  • Use bar charts or written reasoning to describe energy changes in ramps, pendulums, or spring systems.
• Common mistakes:
  • Applying mechanical energy conservation even when friction does significant work.
  • Forgetting to include spring potential energy when a spring is involved.
  • Mixing up the system: including Earth sometimes (for gravity) but not being consistent with where U_g “lives.”

Nonconservative Work and the General Energy Equation

Why you need a more general energy model

In the real world, forces like kinetic friction, air resistance, and some applied forces add or remove mechanical energy. Mechanical energy is no longer conserved, but total energy is still conserved. In AP Physics 1, you typically handle this by adding terms for work done by external forces and/or energy converted to thermal.

A very common and useful relationship is:

W_{\text{nc}} = \Delta E_{\text{mech}}

where

\Delta E_{\text{mech}} = (K_f + U_f) - (K_i + U_i)

Here, W_{\text{nc}} is the work done by nonconservative forces (like friction) on the system.

• If W_{\text{nc}} < 0, mechanical energy decreases (often turning into thermal energy).
• If W_{\text{nc}} > 0, mechanical energy increases (for example, an external motor doing work on the system).

Friction as an energy transfer pathway

Kinetic friction typically does negative work on a moving object because it points opposite displacement:

W_f = -f_k d

(That expression assumes friction is directly opposite motion.)

The “lost” mechanical energy is not lost; it becomes thermal energy in the surfaces. Many AP problems model this as an increase in thermal energy \Delta E_{\text{th}}.

A common representation is:

E_i + W_{\text{ext}} = E_f

where E can include kinetic, potential, and thermal energies, depending on what your system includes.

Choosing a system: the hidden key to energy problems

Energy methods become much easier when you are clear about the system boundary.

Two common choices:

1. Object-only system: Then forces from outside (gravity, friction, applied forces) do work on the system. This approach aligns with W_{\text{net}} = \Delta K.
2. Object + Earth (and maybe spring) system: Then gravity and spring are internal, showing up as potential energy changes. External work might include friction (if the surface is outside the system) or an applied push.

Neither is “the one right system.” The best choice is the one that minimizes bookkeeping.

Example 1: ramp with friction (finding speed)

A block of mass m = 1.0\ \text{kg} slides down a ramp starting from rest at vertical drop h = 1.5\ \text{m}. The path length along the ramp is d = 4.0\ \text{m}. Kinetic friction is constant with magnitude f_k = 2.0\ \text{N}. Find the speed at the bottom.

Choose system: block + Earth, so gravity becomes U_g. Friction is nonconservative and does work.

Initial:

• K_i = 0
• U_{g,i} = mgh

Final:

• U_{g,f} = 0
• K_f = \frac{1}{2}mv^2

Work by friction:

W_f = -f_k d = -(2.0)(4.0) = -8.0\ \text{J}

Use W_{\text{nc}} = \Delta E_{\text{mech}}:

W_f = (K_f + U_{g,f}) - (K_i + U_{g,i})
-8.0 = \left(\frac{1}{2}mv^2 + 0\right) - \left(0 + mgh\right)

Plug in values:

-8.0 = \frac{1}{2}(1.0)v^2 - (1.0)(9.8)(1.5)
-8.0 = \frac{1}{2}v^2 - 14.7
\frac{1}{2}v^2 = 6.7
v^2 = 13.4
v \approx 3.66\ \text{m/s}

Notice how friction reduced the final speed compared with the frictionless result.

Example 2: stopping distance with work-energy

A car of mass m = 1200\ \text{kg} moves at speed v_i = 20\ \text{m/s} and skids to a stop due to kinetic friction. The friction force magnitude is f_k = 6000\ \text{N}. Find the stopping distance d.

Here the object-only model is simplest: the net work done equals change in kinetic energy. Friction does negative work.

Work by friction:

W_f = -f_k d

Change in kinetic energy:

\Delta K = 0 - \frac{1}{2}mv_i^2

Work-energy:

-f_k d = -\frac{1}{2}mv_i^2

Solve:

d = \frac{\frac{1}{2}mv_i^2}{f_k}
d = \frac{\frac{1}{2}(1200)(20)^2}{6000}
d = \frac{600\cdot 400}{6000}
d = 40\ \text{m}

This problem illustrates why energy methods are so efficient: you didn’t need time or acceleration.

Connecting back to “work by gravity” vs. “potential energy change”

A very common confusion is to simultaneously include W_g and \Delta U_g in the same equation, effectively double-counting gravity.

• If you treat gravity as doing work (object-only system), you use W_g.
• If you treat gravity as potential energy (object + Earth system), you use \Delta U_g.

Pick one approach for a given equation.

Exam Focus

• Typical question patterns:
  • Friction or drag present: compute energy “lost” to thermal and find final speed or distance.
  • Mixed representation: energy bar charts showing some energy converted to thermal.
  • Decide what happens if friction increases or decreases (comparative reasoning).
• Common mistakes:
  • Double-counting gravity (including both W_g and \Delta U_g).
  • Using the ramp length for \Delta y (height change is vertical, not along the incline).
  • Forgetting that friction work depends on path length (distance along the surface), not just height.

Power: How Fast Energy Transfers

What power measures

Power is the rate at which energy is transferred or transformed. Even if two people do the same amount of work, the one who does it in less time has greater power.

Average power is

P_{\text{avg}} = \frac{W}{\Delta t}

Instantaneous power (for a force acting on a moving object) is

P = Fv\cos\theta

• P is power in watts (joules per second).
• v is speed.
• \theta is the angle between the force and the velocity.

This formula matches the idea “only the component of force along motion transfers energy.”

Why power shows up in AP Physics 1

Power connects energy ideas to real machines: engines, motors, athletes, elevators, and ramps. It also gives a different way to think about motion: at a given speed, a more powerful engine can supply energy faster.

Efficiency (qualitative emphasis)

In real devices, not all input energy becomes useful output. Efficiency compares useful output to total input, often expressed as a fraction or percent:

\text{efficiency} = \frac{\text{useful output energy}}{\text{input energy}}

You can write a similar ratio with power:

\text{efficiency} = \frac{\text{useful output power}}{\text{input power}}

Efficiency is less than or equal to 1 (or 100%). Energy not in the useful output often becomes thermal due to friction, electrical resistance, sound, or deformation.

Example 1: lifting with power

A motor lifts a m = 50\ \text{kg} load upward at constant speed v = 0.40\ \text{m/s}. Find the power output (useful mechanical power).

At constant speed, the upward force from the cable equals weight:

F = mg

Power:

P = Fv\cos 0^\circ = mgv

Compute:

P = (50)(9.8)(0.40) = 196\ \text{W}

This shows a common AP idea: constant-speed lifting is an energy transfer problem (increasing gravitational potential energy at a steady rate).

Example 2: same work, different power

Two students each climb a vertical height \Delta y = 3.0\ \text{m} on stairs. Student A (mass 60\ \text{kg}) takes 6\ \text{s}; Student B (mass 60\ \text{kg}) takes 3\ \text{s}.

Each does the same work against gravity:

W = mg\Delta y = (60)(9.8)(3.0) = 1764\ \text{J}

Powers:

P_A = \frac{1764}{6} = 294\ \text{W}

P_B = \frac{1764}{3} = 588\ \text{W}

Student B is twice as powerful (same energy transfer in half the time).

Exam Focus

• Typical question patterns:
  • Compute power from work over time (stairs, ramps, motors) or from Fv.
  • Concept questions comparing power vs. work (same work but different times).
  • Efficiency questions interpreting where “lost” energy goes.
• Common mistakes:
  • Confusing work and power (work is energy transferred; power is rate).
  • Using mass times speed instead of weight times speed for vertical lifting.
  • Forgetting the cosine factor in P = Fv\cos\theta when force is not aligned with motion.

Problem-Solving Representations: Energy Diagrams, Graphs, and Strategy Choices

Why AP Physics 1 cares about representations

AP questions often assess not just whether you can compute a number, but whether you can choose and connect representations: equations, graphs, diagrams, and verbal reasoning. Unit 3 is especially representation-heavy because energy is a bookkeeping system.

Free-body diagrams still matter in an “energy unit”

Even when you plan to use energy, a quick free-body diagram helps you decide:

• Which forces do work (parallel components)
• Whether friction is present and its direction
• Whether mechanical energy can be conserved

For example, on an incline, your free-body diagram helps you recognize that gravity has a component along the incline and friction (if present) opposes motion.

Force–position graphs and variable forces

When a force changes with position, work is the area under the F vs. x graph. Springs are the classic case: the spring force increases linearly with x.

If the spring force magnitude is F = kx from x = 0 to x = x_{\max}, then the work done by an external agent compressing it slowly equals the area of a triangle:

W = \frac{1}{2}(\text{base})(\text{height}) = \frac{1}{2}x_{\max}(kx_{\max}) = \frac{1}{2}kx_{\max}^2

That matches the elastic potential energy stored. This connection is valuable: area under the curve is the “graph version” of energy storage in springs.

Energy bar charts: avoiding algebra while staying rigorous

Energy bar charts are not just pictures; they encode conservation and transfer. A disciplined way to use them:

• Decide the system.
• List energy storage forms at initial and final states.
• Add a transfer mechanism if energy enters/leaves (external work) or converts to thermal.

If the final kinetic energy bar is larger, you know net energy into kinetic increased; if thermal appears, you know nonconservative processes occurred.

Choosing between Newton’s laws and energy

A strong AP Physics 1 skill is choosing the most efficient tool.

Energy methods tend to be best when:

• You care about speed after moving through a distance
• Forces vary but are conservative (springs, gravity)
• You want to avoid time

Newton’s second law tends to be best when:

• You care about acceleration at a particular moment
• You need tension/normal forces explicitly
• Motion is constrained and forces determine dynamics (sometimes combined with energy)

Often, the cleanest solution is hybrid:

• Use Newton’s laws to find a force (like friction magnitude f_k = \mu_k N when given coefficients in other units), then use energy to connect states.

Multi-part AP-style example: comparing two paths

Two frictionless tracks start at the same height and end at the same lower height. Track A is steep then flat; Track B is gentle the whole way. Which gives a greater final speed?

Energy reasoning (system: object + Earth, no nonconservative work):

K_i + U_i = K_f + U_f

The final speed depends only on the change in gravitational potential energy mg\Delta y, not the path. Since both tracks have the same vertical drop, both give the same final kinetic energy and thus the same final speed.

A common mistake is to assume the steeper track “must be faster” at the end because it feels more intense early on. It may be faster earlier, but at the end the energy outcome is determined by height change alone when friction is absent.

Notation reference (common symbols you’ll see)

Exam Focus

• Typical question patterns:
  • Use an energy bar chart to justify whether speed increases, decreases, or stays the same.
  • Find work from a piecewise linear F vs. x graph using geometry.
  • Explain why path does or does not matter (conservative vs. nonconservative forces).
• Common mistakes:
  • Treating the area under an F vs. x graph as F/x or slope (it is area, not slope).
  • Saying “energy is conserved” without specifying the system or accounting for friction/thermal.
  • Mixing representations inconsistently (for example, claiming friction is present but drawing no thermal energy increase).