System Energy and Work Models

Driving question

When is an energy model more useful than a force-and-motion model?

Hook: the same hill, two solution paths

A skateboarder starts from rest at the top of a smooth ramp. You could track forces and acceleration at every location. Or you could compare the energy at the top and bottom. The second path often ignores the complicated details between those states.

Energy reasoning follows a compact chain:

choose the system → choose initial and final states → inventory energy stores → identify transfers → test the accounting.

Video learning path

Use the short video as a retrieval tool, not as a replacement for the lesson.

Before watching: Draw a boundary around a cart, ramp, and Earth. Predict what happens to gravitational potential, kinetic, and thermal energy as the cart descends a rough ramp.

During watching: Pause when the 60-joule energy bar appears. Before the split is shown, decide how much kinetic energy remains if $15\text{ J}$ becomes thermal energy. At the system-choice prompt, explain why gravity is external work for a cart-only system but an internal potential-energy interaction for a cart–Earth system.

After watching: Recreate the energy bars without looking. Then solve $45\text{ J}=\frac12(2.0\text{ kg})v^2$ and check whether the result has reasonable units and magnitude. Use the accessible transcript if video or audio is unavailable.

Start with the system boundary

Energy belongs to a defined system, not to a diagram by itself. If the system is a cart alone, Earth's gravitational force can transfer energy by doing external work. If the system is cart plus Earth, gravitational potential energy is an internal store. Both descriptions can be correct; mixing them double-counts energy.

The general accounting model is

where $E$ includes the energy stores chosen for the system and $W_{\text{ext}}$ is energy transferred across the boundary by external work. Thermal transfer or other mechanisms may be named separately when useful.

The chart's equivalent statement is: for a cart–Earth system on a nearly frictionless track, gravitational potential energy decreases while kinetic energy increases by the same amount.

Work transfers energy

For a constant force acting through displacement $\vec d$,

Work is positive when the force component points with displacement, negative when it points opposite displacement, and zero when it is perpendicular. A force can act yet do zero work; for ideal uniform circular motion, the radial force is perpendicular to instantaneous displacement.

The net work on a particle-like object equals its change in kinetic energy:

Worked example: warehouse cart

A $20.0\text{ kg}$ cart moves at $2.00\text{ m/s}$. A worker does $180\text{ J}$ of work on it while friction does $-60.0\text{ J}$. Find its final speed.

The net work is $120\text{ J}$. Initially,

Thus $K_f=160\text{ J}$, so

The worker's work is not the kinetic-energy change because friction also transfers energy.

Potential energy is a system store

Near Earth's surface, the change in gravitational potential energy of an object–Earth system is

For an ideal spring in the Hooke's-law range,

Only changes in potential energy matter, so the zero level is a convenient choice. The sign of $\Delta U_g$ depends on the change in height, not on the direction the object happens to move horizontally.

Energy conservation with dissipation

Energy is not “used up.” It transfers between objects or transforms among stores. If friction acts inside the chosen system, mechanical energy may become thermal energy:

Equivalent text: a decrease in mechanical energy equals the increase in thermal energy when no energy crosses the system boundary by another pathway.

Worked example: braking bicycle

A bicycle and rider of total mass $80.0\text{ kg}$ slow from $10.0\text{ m/s}$ to rest on level ground. For the bicycle–rider–road system, the increase in thermal energy is

The energy has not disappeared. It is distributed mainly as thermal energy in brakes, tires, road, and surroundings.

Power measures transfer rate

Average power is

For a force applied to an object moving with instantaneous velocity,

Power and energy are not interchangeable. Two motors may transfer the same energy, while the more powerful motor does so in less time.

Reading an energy-position graph

For a conservative interaction, force relates to the slope of a potential-energy graph:

Where $U(x)$ rises to the right, force points left. A stable equilibrium occurs at a local minimum of $U$; small displacements produce a force back toward equilibrium.

Equivalent text: the graph has a valley at point A, where nearby slopes direct force back toward A, and a hilltop at point B, where nearby forces direct the object away from B.

Choosing between energy and dynamics

Use an energy model when the question compares states, path details are unnecessary, or force varies with position. Use Newton's laws when you need acceleration, time, direction changes, or individual interaction forces. Sometimes the strongest solution combines them: energy finds speed, then dynamics finds a force.

AP-style evidence routine

1. Name the system and interval.
2. Sketch initial and final states.
3. Inventory $K$, $U_g$, $U_s$, and thermal energy as relevant.
4. Identify transfers across the boundary and assign signs.
5. Write the symbolic accounting equation before numbers.
6. Check joules, signs, plausibility, and limiting cases.
7. Explain why energy or dynamics is the more efficient model.

Misconception clinic

“Energy is consumed.” Energy transfers or transforms; useful mechanical energy may spread into thermal stores.

“Zero net work means no forces act.” Multiple forces may do work that sums to zero, or a force may be perpendicular to displacement.

“Potential energy belongs to one object.” It is associated with an interaction inside a chosen multi-object system.

“Power is energy.” Power is the rate at which energy is transferred.

Retrieval pause

1. Why must the system be named before using potential energy?
2. What sign does friction's work have on a sliding object when the object alone is the system?
3. Can net work be zero while an object moves? Explain.
4. What does a negative slope on a $U(x)$ graph imply about $F_x$?
5. When would Newton's second law be more useful than energy conservation?

Key takeaway

Energy accounting connects states without reconstructing every instant. Its reliability depends on a clear system boundary, explicit transfers, and consistent treatment of potential and thermal energy.

Further learning and alignment

• College Board, “AP Physics 1: Algebra-Based Course,” https://apcentral.collegeboard.org/courses/ap-physics-1 (retrieved 2026-08-18). Standards and course-alignment reference only; no endorsement implied.
• OpenStax, “Work, Energy, and Power,” https://openstax.org/books/physics/pages/9-introduction (retrieved 2026-08-18). CC BY; used for concept verification, not copied exercises.
• The Physics Classroom, “Work, Energy, and Power,” https://www.physicsclassroom.com/class/energy (retrieved 2026-08-18). Concept reference; no prose or exercises reproduced.

Lesson resources

• Book section: Energy Accounting Across a System Boundary
• Supervised investigation: Track Energy Through a Ramp System
• Accessible video transcript
• Energy bar chart
• Friction energy-flow diagram
• Potential-energy landscape