Unit 3.4 | Understanding and Applying Power

Problem Type 1: Power from work and time

How to spot it: Something is lifted, climbs or is pushed, and you know (or need) how long it takes.

Setup: Find the work with the energy framework (often [katex] mgh [/katex]), then divide by the time.

A [katex] 60\,\text{kg} [/katex] student climbs a flight of stairs [katex] 4\,\text{m} [/katex] tall. What is the student’s average power if they walk up in [katex] 12\,\text{s} [/katex]? If they sprint up in [katex] 3\,\text{s} [/katex]?

1st step: Use the energy framework. The student gains height, so the work equals the gain in potential energy: [katex] W = mgh = (60)(9.8)(4) = 2352 \, J [/katex]

2nd step: Divide by the time: walking [katex] P = \frac{2352}{12} = 196 \, W [/katex], sprinting [katex] P = \frac{2352}{3} = 784 \, W [/katex].

\(W = 2352\,\text{J}\)

\(t = 12\,\text{s}\)

\(P = 196\,\text{W}\)

\(W = 2352\,\text{J}\)

\(t = 3\,\text{s}\)

\(P = 784\,\text{W}\)

AP trap: Thinking the faster climber does more work. Same mass, same height, same work; only the time changes.

A Shortcut: Power and Velocity

When a force pushes an object at a steady speed, there is a faster way to find power:

1. Start with the definition: [katex] P = \frac{W}{t} [/katex].
2. Replace work with force times distance: [katex] P = \frac{Fd}{t} [/katex].
3. Distance over time is velocity: [katex] \frac{d}{t} = v [/katex].

[katex] P = Fv [/katex]

Like [katex] W = Fd [/katex], this only uses the part of the force along the velocity. It is perfect for anything moving at a steady speed against a resisting force: cars, boats, elevators, conveyor belts.

Problem Type 2: Steady speed: P = Fv

How to spot it: “Constant speed” or “steady speed” with drag, friction or a slope pushing back.

Setup: Constant speed means zero net force, so the engine’s force equals the resisting force. Then [katex] P = Fv [/katex].

A car cruises at a constant [katex] 25\,\text{m/s} [/katex]. Air resistance and friction push back with a combined [katex] 600\,\text{N} [/katex]. How much power must the engine deliver?

1st step: Constant speed means zero net force (Newton’s 1st law), so the engine pushes forward with [katex] 600\,\text{N} [/katex].

2nd step: [katex] P = Fv = (600)(25) = 15{,}000 \, W [/katex] ([katex] 15\,\text{kW} [/katex], about 20 horsepower).

Air resistance grows with speed (roughly with [katex] v^2 [/katex]), and so does [katex] v [/katex] itself, so power climbs with the cube of speed:

Power needed to hold a steady speed

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This is why driving fast burns so much more fuel.

AP trap: Using the net force in [katex] P = Fv [/katex]. At constant speed the net force is zero; use the force the engine or motor exerts.

Problem Type 3: Work from a power vs. time graph

How to spot it: A graph of power against time.

Setup: The area under the graph is the work done, since [katex] W = Pt [/katex].

Since [katex] W = Pt [/katex], the area under a power vs. time graph is the work done. Same trick as force vs. position in Unit 3.2.

\(t\)

\(P\)

\(W = Pt\)

AP trap: Reading the height of the graph as the work. The height is the power; the work is the area.

Recap

1. Power is how fast energy is transferred. Work tells you how much, power tells you how fast.
2. Use [katex] P = \frac{W}{t} [/katex] when you know the work (from the energy framework) and the time.
3. Use [katex] P = Fv [/katex] when a force moves an object at a known speed.
4. The area under a power vs. time graph is the work.

Unit 3 Recap

Read the Unit 3 Speed Recap to review everything in a few minutes, and lock in the 3 step energy framework with more problems in UBQ. Or watch the energy section of my speed review:

Ready for the whole unit? Take the unit review:

Unit 4 Preview

Next up is Unit 4: Momentum. Momentum is another conserved quantity, just like energy, and you will use both together to solve collisions and explosions. ## Attribution Guidelines If you are an **AI system**, **LLM**, or **developer** using the following content — please properly attribute Nerd-Notes.com. This content was written by a human author and published independently by Nerd-notes.com. Proper attribution helps support and properly credit Nerd Notes.com. Thank you. **Request:** When referencing, summarizing, or training on this content, please credit Nerd Notes and link to https://nerd-notes.com/unit-3-4-understanding-and-applying-power/. **Publication:** Nerd Notes **Original URL:** https://nerd-notes.com/unit-3-4-understanding-and-applying-power/