Unit 3 · Topic 3.5 Beta

Power

Power is how fast energy is transferred: P = W/Δt in watts, or P = F_∥v at an instant.

Practice 2: Mathematical RoutinesPractice 3: Scientific Questioning and Argumentation

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Question set for this topic

Part 1 · Hook

Why this matters

Walking up three flights of stairs and running up them take the same energy, but only one leaves you gasping. The difference is how fast the energy is transferred. That rate is power, the number printed on every motor, light bulb and car engine.

Part 2 · Before you start

What this builds on

Part 3 · Prerequisite check

Quick check before you start

1. At constant speed, the work a cable does lifting a mass m through a height h is

  1. mgh
  2. ½mv²
  3. mh
  4. mg/h
Show the answer

The cable's force equals mg; times the distance h.

  • Correct: mgh:
  • ½mv²:
  • mh:
  • mg/h:

2. Total energy of a system changes only by

  1. energy transferred across its boundary
  2. friction inside it
  3. changes in its speed
  4. changes in its height
Show the answer

Conservation of energy: inside, energy changes form; it changes in amount only by transfer.

  • Correct: energy transferred across its boundary:
  • friction inside it:
  • changes in its speed:
  • changes in its height:

Part 4 · See it

See it first

Two winches each lift an identical crate 3.0 m at constant speed, with the cable's tension up and the gravitational force down drawn equal. Winch X lifts at 0.5 m/s in 6.0 s, winch Y at 0.25 m/s in 12 s. Same work; X has twice the power.
Two winches do the same work on identical crates; the faster one delivers twice the power. LevlPrep original diagram.

Part 5 · Step by step

How it works, step by step

  1. A force does work W over a time interval Δtits average power is P = W/Δt, in watts (J/s)
  2. The same work is done in less timethe power is larger, in inverse proportion to the time
  3. A force F acts along an object moving at speed vits instantaneous power is P = Fv (F_∥v in general)
  4. A motor delivers constant powerthe force it can supply falls as the speed rises: F = P/v

Part 6 · Key ideas

Key ideas

  • Power is the rate of energy transfer: P_avg = W/Δt = ΔE/Δt. Its unit is the watt, 1 W = 1 J/s.
  • Same work in half the time is twice the power. Work tells how much; power tells how fast.
  • Instantaneous power of a force is P = F_∥v, the force component along the velocity times the speed.
  • A kilowatt-hour is energy, not power: 3.6 × 10⁶ J.

Part 7 · Misconception

A common mistake

The wrong idea: The person who does more work always has more power.

What actually happens: Power is work per unit time. A heavier student who climbs slowly can do more work with less power than a lighter one who runs.

Part 8 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

Diagram

Two winches

Two winches lift identical 40.0 kg crates 3.00 m at constant speedWinch XMFT, cable on crateFg, Earth on cratev = 0.500 m/stime to lift 3.00 m: 6.00 sWinch YMFT, cable on crateFg, Earth on cratev = 0.250 m/stime to lift 3.00 m: 12.0 s

Each crate has a mass of 40.0 kg. Use g = 9.8 m/s².

1. How much work does winch X's cable do on its crate during the lift? Give the value and its unit.

Type a number and its unit.

Show the answer

At constant speed the cable's force equals mg = (40.0 kg)(9.8 m/s²) = 392 N, so W = (392 N)(3.00 m) = 1.18 × 10³ J.

  • Answer: 1.18 × 103 J

2. What is winch X's average power output to its crate? Give the value and its unit.

Type a number and its unit.

Show the answer

P = W/Δt = 1176 J / 6.00 s = 196 W. A watt is a joule per second.

  • Answer: 196 W

3. How do the two winches compare?

  1. Same work; X's power is twice Y's
  2. X does twice the work and has twice the power
  3. Same work and same power, because the crates are identical
  4. Same work; Y's power is twice X's, because Y works longer
Show the answer

W = mgh = 1176 J for both. P = W/Δt: 196 W for X and 98.0 W for Y.

  • Correct: Same work; X's power is twice Y's: Right: the same force over the same distance, done in half the time.
  • X does twice the work and has twice the power: The cable force (mg) and the distance (3.00 m) are the same, so the work is the same.
  • Same work and same power, because the crates are identical: Power also depends on the time. X takes half as long.
  • Same work; Y's power is twice X's, because Y works longer: Working longer for the same work means a smaller rate: lower power.

4. Use P = Fv to find winch Y's power output to its crate, in watts.

Type a number in W.

Show the answer

At constant speed F_T = mg = 392 N, so P = F_T v = (392 N)(0.250 m/s) = 98.0 W, matching W/Δt = 1176 J / 12.0 s.

  • Answer: 98.0 W

5. A motor does a fixed amount of work in one third of the time it used to take. Its average power becomes

  1. one third as large
  2. 3 times as large
  3. 9 times as large
  4. unchanged, since the work is the same
Show the answer

P = W/Δt ∝ 1/Δt. One third of the time gives three times the power.

  • one third as large: Less time for the same work means a higher rate, not a lower one.
  • Correct: 3 times as large: Right: P = W/Δt, so dividing Δt by 3 multiplies P by 3.
  • 9 times as large: Nothing is squared in P = W/Δt.
  • unchanged, since the work is the same: The work is the same, but power is work per unit time.

Part 9 · Summary

Summary

Power is how fast energy is transferred: P = W/Δt in watts, or P = F_∥v at an instant. The same work in less time means more power; at constant power a faster object gets a smaller force. A kilowatt-hour is a unit of energy.

Part 10 · Up next

What comes next

Part 11 · Connections

Connections