AP Physics 1 · Unit 3 · Topic 3.5
Power
Power describes how quickly energy is transferred into, out of, or within a system. Two processes can transfer the same energy yet have very different power because they take different amounts of time.
Learning Goals
After this lesson, you should be able to:
- distinguish energy from power and use correct SI units;
- calculate average power from work or energy change;
- derive and apply \(P=Fv\cos\theta\);
- determine whether a force delivers positive, negative, or zero power;
- interpret energy–time and power–time graphs;
- compare constant-force and constant-power motion;
- analyze useful output and efficiency; and
- design a measurement of human mechanical power.
1. Average Power
Average power is the energy transferred or converted during a time interval divided by that interval:
Use \(W/\Delta t\) when a force's work is known, and \(\Delta E/\Delta t\) when an energy ledger is easier. Power does not tell how much energy is available unless the operating time is also known.
2. Energy and Power Are Different
| Quantity | Meaning | SI unit | Example question |
|---|---|---|---|
| Energy \(E\) | Amount stored or transferred | joule, \(\mathrm J\) | How much energy changed? |
| Power \(P\) | Rate of energy change | watt, \(\mathrm W\) | How quickly did it change? |
A \(500\,\mathrm W\) device transfers \(500\,\mathrm J\) each second while operating at that power.
3. Same Energy, Different Rate
4. Worked Example: Running Upstairs
Question: A \(65\,\mathrm{kg}\) student runs up stairs with a vertical rise of \(3.2\,\mathrm m\) in \(4.0\,\mathrm s\). The student's starting and ending speeds are approximately equal. Find the average mechanical power.
Answer: The average mechanical power is about \(510\,\mathrm W\). This is not the person's total metabolic power because biological efficiency and other energy transfers are not included.
5. Instantaneous Power from Force and Velocity
During a short interval, a constant force does work \(W=F\Delta r\cos\theta\). Dividing by the interval and using \(v=\Delta r/\Delta t\) gives
Here \(\theta\) is the angle between the force and velocity. Only the force component parallel to the instantaneous velocity transfers energy at that moment.
6. The Sign of Power
| Force–velocity angle | Power | Effect of that force |
|---|---|---|
| \(0^\circ\le\theta<90^\circ\) | Positive | Transfers energy into the object's motion or selected system |
| \(\theta=90^\circ\) | Zero | Changes direction without changing kinetic energy |
| \(90^\circ<\theta\le180^\circ\) | Negative | Transfers energy out of the object's motion or selected system |
Power is signed because it is a transfer rate. A negative power value does not mean a device has “negative capacity”; it identifies the direction of energy transfer under the chosen convention.
7. Worked Example: Angled Towing Force
Question: A tow cable exerts a constant \(1.20\times10^3\,\mathrm N\) force at \(20^\circ\) to a vehicle's velocity. At an instant when the speed is \(12.0\,\mathrm{m/s}\), find the cable's delivered power.
Answer: The cable delivers about \(13.5\,\mathrm{kW}\). Using \(Fv\) without the cosine would incorrectly count the perpendicular force component.
8. Net Power and Kinetic-Energy Change
The average net power delivered to an object equals its kinetic-energy change per time:
Individual forces can simultaneously deliver positive, negative, or zero power. Their signed powers add to the net power, just as their signed works add to net work.
9. Worked Example: Accelerating Vehicle
Question: A \(900\,\mathrm{kg}\) vehicle speeds up from \(10.0\) to \(20.0\,\mathrm{m/s}\) in \(8.0\,\mathrm s\) on level ground. Find the average net power associated with the kinetic-energy change.
Answer: The average net power is \(16.9\,\mathrm{kW}\). An engine's actual output must be larger if drag and rolling resistance also receive energy.
10. Constant Force Is Not Constant Power
If a constant force remains parallel to motion, \(P=Fv\). As the object speeds up, the same force delivers increasing instantaneous power.
If a motor instead provides constant power, the available forward force satisfies
At greater speed the same constant power corresponds to less force. This distinction is essential when interpreting engine ratings and motion graphs.
11. Reading Energy–Time and Power–Time Graphs
- The slope of an \(E\)-versus-\(t\) graph represents power.
- A steeper positive slope means faster energy transfer into the modeled store.
- A horizontal energy graph means zero net power for that store.
- Signed area under a \(P\)-versus-\(t\) graph represents energy transfer.
- Regions below the time axis subtract from the net transferred energy.
For algebra-based problems, add rectangle, triangle, or trapezoid areas rather than requiring calculus.
12. Worked Example: Area Under a Power Graph
Question: Power increases linearly from \(0\) to \(600\,\mathrm W\) during the first \(4.0\,\mathrm s\), then remains at \(600\,\mathrm W\) for \(3.0\,\mathrm s\). How much energy is transferred?
Answer: The total transfer is \(3.0\,\mathrm{kJ}\). Multiplying the final \(600\,\mathrm W\) by all \(7.0\,\mathrm s\) would ignore the initial ramp.
13. Useful Output and Efficiency
When an input power is divided among useful output and other transfers,
Example: An elevator needs \(9.8\,\mathrm{kW}\) of useful mechanical power while rising steadily. If the motor system is \(80\%\) efficient,
Efficiency is dimensionless and cannot exceed \(1\) or \(100\%\) for a correctly defined passive conversion process.
14. Investigation: Measure Stair-Climbing Power
Use a staircase, meterstick, balance, stopwatch or video, and several participants.
- Measure each participant's mass and the total vertical rise.
- Define a consistent start and finish line.
- Record several ascent times while maintaining safe movement.
- Calculate \(m g\Delta y\) and \(P_{\mathrm{avg}}=m g\Delta y/\Delta t\).
- Compare repeated trials and graph power against mass, time, or climbing rate.
- Estimate timing and height uncertainties and explain why mechanical power differs from metabolic power.
Do not compare participants using stair-path length in place of vertical rise.
15. Common Traps and Key Takeaways
- Power is a rate, not an amount of energy.
- Average and instantaneous power are not automatically equal.
- Use the force component parallel to velocity in \(P=Fv\cos\theta\).
- Keep watts and joules distinct: multiply power by time to obtain energy.
- Power may be negative or zero even when a force is nonzero.
- On a \(P\)-versus-\(t\) graph, use signed area; on an \(E\)-versus-\(t\) graph, use slope.
- Constant force and constant power predict different behavior as speed changes.
A \(75\,\mathrm{kg}\) skier is pulled uphill through a vertical rise of \(6.0\,\mathrm m\) in \(12.0\,\mathrm s\). The skier's speed increases from \(2.0\) to \(4.0\,\mathrm{m/s}\), and \(1.20\,\mathrm{kJ}\) of mechanical energy is converted to thermal energy. Determine the average input power. Draw an energy-transfer diagram and identify which terms would disappear if the initial and final speeds were equal.
Concept scope cross-checked with the College Board AP Physics 1 Course and Exam Description and the open educational references OpenStax University Physics: Power and OpenStax College Physics for AP Courses: Power. Explanations, diagram, examples, checkpoint, and investigation are independently written for Insomnia Studyroom.
Official curriculum reference: College Board AP Physics 1 course page. The explanation and worked example are independently written for this study site.