Net work changes kinetic energy
You will be able to: Derive and use W_net = ΔK while respecting stopping points.
How does Newton’s second law become an energy equation?
A 2 kg cart moves at 3 m/s, carrying 9 J of kinetic energy. If all forces together do another 7 J of work, it has 16 J and moves at 4 m/s. We can find its speed without first finding travel time.
A useful starting point: Work is signed area, even when force changes →
Words and symbols before equations
- Net work W_net
- Sum of work by every force on the chosen object.
- ΔK
- Final kinetic energy minus initial kinetic energy.
- Chain rule
- For one-dimensional motion, a = dv/dt = v dv/dx along a smooth segment.
- Turning or stopping point
- A location where speed reaches zero; continuing motion needs a new direction or a new model.
What this picture assumes
Fixed level track, point object, constant forces during rightward motion. Weight and normal do no work. At an unreachable endpoint, the model reports the first stop and does not extend the assumed rightward slide.
Read the picture in three steps
- Locate the labeled sources, system boundary or graph axes. Read the units before comparing values.
- Initial K 16 J. At 3 m: K = 22 J; speed 4.69 m/s. Net force 2 N.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the physics
For a fixed-mass particle, F_net = ma. Multiply by dx and use a dx = v dv: F_net dx = mv dv. Integrating gives ∫F_net dx = ∫mv dv = ½mv_f² − ½mv_i². This is the work–energy theorem; the vector derivation gives the same result in multiple dimensions.
Add work from every force. Positive applied work does not guarantee increasing K if friction removes still more energy. Energy predicts speed, not its direction; signs or a motion diagram supply direction.
The theorem applies to translational point-object models here. A negative computed K is not a physical result: it means the object cannot reach the stated point under the assumed motion. For sliding friction, its direction reverses with sliding direction, so do not extend a rightward-friction formula through a reversal.
A worked example, step by step
A 2 kg cart starts at 4 m/s. Over a 3 m rightward displacement, a 5 N pull acts right and a 3 N resistance acts left. Find its final speed.
- Choose the cart and the lab frame. Weight and normal do zero work on this horizontal path.
- Initial K = ½(2)(4²) = 16 J. Pull work is 15 J and resistance work is −9 J.
- Net work is 6 J, so K_f = 16 + 6 = 22 J.
- v_f = √(2K_f/m) = √22 = 4.69 m/s. It speeds up because net work, rather than pull work alone, is positive.
W_net equals the change in K, not K itself unless the object starts from rest.
Can zero net work occur while several forces do work?
Compare with an explanation
Yes. Their positive and negative work can cancel, leaving speed unchanged.
Predict. Change one thing. Explain.
Set initial speed to 4 m/s, pull to 0 N and resistance to 4 N. Find where the cart stops. Increase the requested distance and read why the model rejects a later endpoint.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Initial K 16 J. At 3 m: K = 22 J; speed 4.69 m/s. Net force 2 N.
Fixed level track, point object, constant forces during rightward motion. Weight and normal do no work. At an unreachable endpoint, the model reports the first stop and does not extend the assumed rightward slide.
Explain what you noticed: Which quantity changed? Which stayed fixed? Use the relevant work, system boundary, energy or calculus relationship to justify your prediction.
Apply the idea to a fresh problem Practice →Show what you understand.
Two original questions are a starting check, not proof of mastery. Explain your choice before revealing the answer.
Original written challenge
4 points · self-check · not an official AP questionA 1 kg block moving at 6 m/s slides on a fixed floor against constant 3 N friction with no other horizontal force. Find stopping distance and explain why a calculation at 8 m fails.
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Compare with the answer and four-point rubric
- 1 point: Initial K = ½(1)(6²) = 18 J.
- 1 point: At the stop, ΔK = −18 J and friction work = −3d.
- 1 point: Solve d = 6 m.
- 1 point: The block never reaches 8 m during this slide; K = 18 − 24 = −6 J flags an impossible assumed endpoint.
Accept equivalent correct methods and explanations. This is a Refresh Kid teaching rubric, not an official AP scoring guideline.
Retrieve it before you reveal it.
RECALL 1What work belongs in ΔK?
Net work, the sum over all forces on the particle.
RECALL 2What does a negative predicted K mean?
The proposed state is inaccessible under the assumptions.
RECALL 3Which calculus step connects force and kinetic energy?
a dx = v dv, giving ∫m v dv = Δ(½mv²).
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Net work changes kinetic energy
- ΣW = ΔK = ½m(v_f² − v_i²).
- For constant net force along x: v_f² = v_i² + 2F_netΔx/m.
- K_f must be nonnegative; check whether an endpoint is reachable.
Remember: W_net equals the change in K, not K itself unless the object starts from rest.
Conditions: Fixed level track, point object, constant forces during rightward motion. Weight and normal do no work. At an unreachable endpoint, the model reports the first stop and does not extend the assumed rightward slide.
Refresh Kid · AP Physics C: Mechanics Unit 3 (official Unit 3) · Objectives 3.2.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 3.2, objectives 3.2.A. CED effective Fall 2024, current PDF ©2026; checked September 16, 2026 alongside the Fall 2026 clarifications. This is Mechanics Unit 3: Work, Energy, and Power. The unit covers Topics 3.1–3.5. Calculus connects work to force integrals, force to potential-energy derivatives, and power to the rate of energy transfer. Models distinguish object-only and multi-object systems; translational models exclude rotational energy unless explicitly noted. The lesson breakdown and questions are original Refresh Kid work, not official topic subdivisions.
Implementation and automated checks are separate from independent teacher review and observation of students. Both human review stages remain pending. This is a review edition, not a certified or validated assessment.
Optional further resource: College Board’s released questions and scoring guides. Papers can combine units; this link is an archive, not an assignment of every question to this lesson.
Our learn, explore, practice and recall sequence is informed by the IES learning guide. The exact Refresh Kid implementation has not been evaluated for learning effectiveness.
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