From net force to a motion function
You will be able to: Apply ΣF = ma and integrate a time-dependent force for constant mass.
How does a changing force build a changing velocity?
A 2 kg cart starts from rest. Its motor supplies a net force F(t) = (4 N/s)t. The force is 4 N at 1 s and 8 N at 2 s, so acceleration is not constant.
A useful starting point: Balanced forces and inertial frames →
Words and symbols before equations
- Inertial mass m
- Resistance to acceleration, measured in kg.
- Newton’s second law
- For constant mass, ΣF = ma = m dv/dt.
- Initial condition
- A known position or velocity at a specified time.
- Net-force graph
- A graph of the total force component, not necessarily one applied force.
What this picture assumes
Constant mass, net horizontal force F = ct, v(0) = 1 m/s, x(0) = 0. The force shown is the total horizontal force. Vertical weight and support balance.
Read the picture in three steps
- Locate the labeled sources, system boundary or graph axes. Read the units before comparing values.
- At 3 s: F_net = 12 N, a = 6 m/s², v = 10 m/s, x = 12 m.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the physics
Choose a system and axis, then add the external force components. With constant mass, a(t) = F_net(t)/m. Net force points along acceleration; it need not point along the current velocity.
For F = ct, integrating dv/dt = ct/m gives v(t) = v₀ + ct²/(2m). Integrating again gives x(t) = x₀ + v₀t + ct³/(6m). Each integration needs its own initial value. This is the same calculus used in Unit 1, now tied to an interaction.
An experiment can test a proportionality by varying net force at fixed total mass and plotting a against F_net. The ideal slope is 1/m. Repeated trials, calibrated force readings and friction estimates matter; an applied-force reading is not automatically the net force.
A worked example, step by step
For a 2 kg cart with F_net(t) = (4 N/s)t, x₀ = 0 and v₀ = 1 m/s, find velocity and position at 3 s.
- a(t) = F/m = (2 m/s³)t.
- Integrate: v(t) = 1 m/s + (1 m/s³)t².
- At 3 s, v = 1 + 9 = 10 m/s.
- Integrate v: x(t) = (1 m/s)t + (1/3 m/s³)t³, so x(3) = 3 + 9 = 12 m.
Use the net force, not whichever force is easiest to identify. A varying force usually means varying acceleration.
Can force point left while an object moves right?
Compare with an explanation
Yes. It then has leftward acceleration and can be slowing down while still moving right.
Predict. Change one thing. Explain.
Hold mass and initial values fixed. Double the force coefficient. Predict which parts of velocity and position double and which initial contributions stay fixed.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
At 3 s: F_net = 12 N, a = 6 m/s², v = 10 m/s, x = 12 m.
Constant mass, net horizontal force F = ct, v(0) = 1 m/s, x(0) = 0. The force shown is the total horizontal force. Vertical weight and support balance.
Explain what you noticed: Which quantity changed? Which stayed fixed? Use the relevant force, system boundary, acceleration 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 questionDesign a test of a = F_net/m at fixed total mass. State the graph, expected slope and a way to address friction. For a slope of 0.25 kg⁻¹, determine mass.
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Compare with the answer and four-point rubric
- 1 point: Vary and measure net force while keeping total moving mass fixed; measure acceleration.
- 1 point: Plot acceleration vertically against net force horizontally.
- 1 point: Expected slope is 1/m, giving m = 4 kg for 0.25 kg⁻¹.
- 1 point: Estimate friction rather than treating the pull as net force; repeat trials and consider uncertainty in the fit.
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 determines acceleration direction?
Net external force direction.
RECALL 2What does integrating F_net/m give?
Velocity change for a constant-mass system.
RECALL 3Why retain initial values?
Force determines changes, not the starting motion.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
From net force to a motion function
- ΣF = ma for fixed mass.
- v(t) = v₀ + ∫F_net(t)/m dt over the stated time interval.
- At fixed mass, an a-versus-F_net graph has slope 1/m.
Remember: Use the net force, not whichever force is easiest to identify. A varying force usually means varying acceleration.
Conditions: Constant mass, net horizontal force F = ct, v(0) = 1 m/s, x(0) = 0. The force shown is the total horizontal force. Vertical weight and support balance.
Refresh Kid · AP Physics C: Mechanics Unit 2 (official Unit 2) · Objectives 2.5.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 2.5, objectives 2.5.A. CED effective Fall 2024, current PDF ©2026; checked September 16, 2026 alongside the Fall 2026 clarifications. This is Mechanics Unit 2: Force and Translational Dynamics. The unit covers Topics 2.1–2.10. Calculus is introduced where it is needed for continuous mass and velocity-dependent forces. Shell theorem is applied without requiring a proof; spring combinations are purely series or purely parallel. 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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