Power at an instant: force and velocity
You will be able to: Use the force component along velocity to calculate instantaneous mechanical power.
Why can a steady pulling force deliver more power as speed rises?
Pull a cart with a steady 10 N force. At 1 m/s, the force transfers 10 J each second. At 3 m/s, its application point covers three times as much distance each second, so the transfer rate is three times larger.
A useful starting point: Average power →
Words and symbols before equations
- Instantaneous power
- Energy-transfer rate at a particular moment, in W.
- Velocity v
- Velocity of the force’s point of application; m/s.
- F_parallel
- Component of force along the instantaneous motion, in N.
What this picture assumes
Constant force magnitude 10 N. Velocity is at its point of application. Diagram shows an instantaneous vector relationship, not a complete force list or trajectory.
Connect the picture to the physics
For a force acting at a moving point, P=Fv cosθ. Force times distance is work; distance per time is speed, so force times speed gives the transfer rate when the directions align.
A force opposing velocity delivers negative power to the object: it removes mechanical energy. A perpendicular force delivers zero instantaneous power even if it changes velocity direction.
Constant force does not imply constant power if speed changes. Conversely, a constant-power drive can supply less forward force at higher speed: F=P/v for aligned force and nonzero speed. That formula cannot be used to claim an infinite real force at zero speed.
A worked example, step by step
A motor exerts 20 N forward on a cart moving at 3 m/s. Resistance is 5 N backward. Find each force’s instantaneous power and their net transfer rate.
- Motor power is 20×3=+60 W.
- Resistance power is −5×3=−15 W.
- Net power is +45 W, the kinetic-energy increase rate for this particle model.
- If the same forces act when speed reaches 6 m/s, all these rates double; the forces themselves did not change.
Use the velocity at the force’s point of application. Do not multiply unrelated force and speed values.
At an instant when the cart is at rest, can a nonzero force have zero instantaneous power?
Compare with an explanation
Yes. P=Fv gives zero at v=0, even though the force can accelerate it and do work over a later interval.
Predict. Change one thing. Explain.
Keep force magnitude 10 N. Change speed and then the angle between force and velocity. Compare positive, zero and negative power at the same speed.
Instantaneous power = 30 W. Force–velocity angle 0°. At zero speed, power is zero.
Constant force magnitude 10 N. Velocity is at its point of application. Diagram shows an instantaneous vector relationship, not a complete force list or trajectory.
Explain what you noticed: Which quantity changed? Which stayed fixed? Use a work, energy or power 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 questionAn ideal drive supplies 120 W of forward mechanical power at a moment when its cart moves at 4 m/s. (a) Find force. (b) Find force at 8 m/s with the same power. (c) Explain the change. (d) State why the formula must not be extrapolated to an infinite force at rest.
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Compare with the answer and four-point rubric
- 1 point: 30 N.
- 1 point: 15 N.
- 1 point: At higher speed, less force transfers the same energy per second.
- 1 point: The ideal constant-power relation does not describe a real drive’s startup; real devices have force and operating limits.
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 1Instantaneous power from force and motion?
The dot product F·v, or Fv cosθ.
RECALL 2Can a nonzero force have zero power?
Yes, if velocity is zero or perpendicular.
RECALL 3Does constant force imply constant power?
No; power also depends on speed and angle.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Power at an instant: force and velocity
- P=Fv cosθ=F_parallel v.
- Aligned nonzero speed: F=P/v.
- Net power equals the kinetic-energy change rate in the particle model.
Remember: Use the velocity at the force’s point of application. Do not multiply unrelated force and speed values.
Conditions: Constant force magnitude 10 N. Velocity is at its point of application. Diagram shows an instantaneous vector relationship, not a complete force list or trajectory.
Refresh Kid · Unit 3 · Objectives 3.5.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 3.5, objectives 3.5.A. CED effective Fall 2024, current PDF ©2026; checked September 16, 2026. Fall-2026 corrections also checked. 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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