D. Energy and Society · Work, Energy and PowerInvestigation: How Powerful Are You?
D2.6 D2.5 A1.1 A1.4 A1.5 A1.6 A1.8 A1.13About 65 minutes
Learning goals- I can plan and safely carry out an investigation to measure my power output climbing stairs.
- I can calculate work and power from measurements of mass, height and time.
- I can estimate the uncertainty in my result and explain why it underestimates the power my body uses.
- I can solve problems linking power, work and time.
Words to know
power output
The rate at which useful work is done, here the rate of gaining gravitational potential energy: P = mgh ÷ Δt.
riser
The vertical part of one stair step. Total height = number of risers × riser height.
reaction time
The delay between seeing something and responding (for example, starting a stopwatch), often about 0.2 s.
uncertainty
The range within which the true value probably lies, such as 3.1 s ± 0.2 s.
metabolic power
The rate at which your body transforms chemical energy from food. It is several times larger than your useful mechanical power output.
Are you a one-horse student?
Engines are rated in kilowatts or horsepower. What about you? In this investigation you will measure your own power output by timing how fast you climb a flight of stairs. You will turn a short burst of effort into a number in watts, and then compare it with a light bulb, a kettle and a horse.
The plan
Question: What is my power output when I climb a flight of stairs briskly, compared with walking slowly?
Physics: Climbing raises your body a height h. The useful work is the gain in gravitational potential energy, W = mgh. Your power is
P = W ÷ Δt = mgh ÷ Δt
Measurements:
- m: your mass, from a bathroom scale (kg)
- h: the vertical height of the stairs. Count the risers (n), measure one riser height (r) and calculate h = n × r. Do not measure along the slope of the stairs.
- Δt: the time from the first step to the top, with a stopwatch
Variables: independent: climbing pace (slow walk or brisk climb); dependent: time and power; controlled: the same person, the same stairs, the same starting position.
Safety (A1.4): climb briskly but safely, one step at a time, holding the handrail. Use dry, well-lit stairs with no objects on them, wear proper shoes, and have an adult present. Do not run if you have asthma, a heart condition or an injury; a slow climb works too.
Worked exampleExample 1: Analysing the sample data
A 58 kg student climbs 14 steps, each 0.18 m high. Three brisk trials take 3.2 s, 3.0 s and 3.1 s. Find her average power, in watts and horsepower.
- Height: h = 14 × 0.18 m = 2.52 m.
- Work: W = mgh = (58)(9.8)(2.52) ≈ 1432 J.
- Average time: (3.2 + 3.0 + 3.1) ÷ 3 = 3.1 s.
- Power: P = 1432 J ÷ 3.1 s ≈ 462 W ≈ 4.6 × 10² W.
- In horsepower: 462 ÷ 746 ≈ 0.62 hp.
About 4.6 × 10² W, or 0.62 hp.
| Trial | Pace | Time (s) | W = mgh (J) | P (W) |
|---|
| 1 | brisk | 3.2 | 1432 | 448 |
| 2 | brisk | 3.0 | 1432 | 477 |
| 3 | brisk | 3.1 | 1432 | 462 |
| 4 | slow walk | 7.5 | 1432 | 191 |
How good is the result?
Timing uncertainty. Starting and stopping a stopwatch by hand adds an error of about ±0.2 s. On a 3.1 s climb that is about ±6%, so the power is roughly 460 W ± 30 W. A longer staircase, or filming the climb and timing from the video, makes the percentage smaller.
Height uncertainty. Measuring one riser to ±0.5 cm and multiplying by 14 gives ±7 cm in h (about ±3%). Measuring several risers and averaging helps.
What we left out (systematic). Our calculation counts only the gain in Eg. It ignores the kinetic energy you have at the top, the energy used to swing your legs and arms, and the fact that muscles are only about 20% to 25% efficient. Your body's metabolic power during the climb was several times larger than 460 W; most of it became thermal energy, which is why you warm up.
Significant digits. With times to 2 significant digits, report about 2 significant digits: 4.6 × 10² W.
Worked exampleExample 2: Step-ups instead of stairs
No stairs? The same student does 20 step-ups onto a sturdy 0.20 m step in 30 s. Find her power output.
- Each step-up raises her 0.20 m: W per step = (58)(9.8)(0.20) = 113.68 J.
- 20 step-ups: W = 20 × 113.68 ≈ 2274 J. (Stepping down does not add to the useful work.)
- P = 2274 J ÷ 30 s ≈ 76 W.
About 76 W. That is far less than the stair sprint, because the pace is steady rather than a burst.
Just for funThe sample student's power was 0.62 hp, so she is officially about three-fifths of a horse. (Please do not add that to a résumé.)
Hands-on activityMeasure your power
You need: a bathroom scale, a tape measure, a stopwatch (or a phone) and a helper, and a flight of indoor stairs with a handrail (or a sturdy, non-slip step for step-ups).
Grown-up note: an adult should be present. Stairs must be dry, well-lit and clear. Climb one step at a time, holding the handrail. Students with asthma, heart conditions or injuries should do the slow-walk version only, or be the timer.
- Measure your mass. Count the risers and measure the height of one riser (or three, and average). Calculate h.
- Your helper says "go" and starts the stopwatch as your foot leaves the floor, and stops it when your foot lands on the top step.
- Do three brisk trials (rest in between) and one slow walk.
- Copy the sample table and fill it in with your data. Calculate your average power for the brisk climb in watts and horsepower.
- Estimate the uncertainty in your time and in your power.
- Write a conclusion that answers the question and gives two reasons why your real metabolic power was greater than the number you calculated.
What to look for: many teens get somewhere in the range of a few hundred watts for a brisk climb. Your result depends on your mass, the stairs and your pace. Higher-mass students are not "less powerful": they do more work on each climb.
Your turn
1A 65 kg student climbs stairs 3.2 m high in 4.0 s. Find his power.
W = (65)(9.8)(3.2) = 2038.4 J. P = 2038.4 ÷ 4.0 ≈ 510 W (5.1 × 10² W).
2Why must you use the vertical height of the stairs, not their slope length?
Gravity is vertical, so the work done against it is mgh with h the vertical rise. Horizontal motion along the stairs does no work against gravity.
3A staircase has 16 risers, each 17.5 cm. What is its vertical height in metres?
h = 16 × 0.175 m = 2.80 m.
4A student's times are 4.1 s, 4.4 s and 4.2 s. What is the average time, and why average?
(4.1 + 4.4 + 4.2) ÷ 3 = 4.2 s (4.23 s). Averaging reduces the effect of random timing errors.
5A 50 kg student's power is 400 W. How long does she take to climb 3.0 m?
W = (50)(9.8)(3.0) = 1470 J. Δt = 1470 ÷ 400 ≈ 3.7 s.
6Give one safety precaution for the stair investigation and explain why it matters.
Sample: hold the handrail and take one step at a time, so you don't trip or fall while hurrying. (Others: dry, clear stairs; proper shoes; an adult present.)
More practice: warm-up
1Write the formula for the power output when climbing.
2Find the work done by a 60 kg person climbing 2.0 m.
(60)(9.8)(2.0) = 1176 J ≈ 1.2 × 10³ J.
3If the same climb takes 4.0 s, what is the power?
4Convert 373 W to horsepower.
5If you climb the same stairs in half the time, what happens to your power?
More practice: on your own
1Two students climb the same 3.0 m stairs in 4.0 s. One has a mass of 50 kg, the other 75 kg. Compare their power outputs.
50 kg: (50)(9.8)(3.0) ÷ 4.0 = 367.5 W. 75 kg: 551.25 W. The 75 kg student has 1.5 times the power, because he does 1.5 times the work in the same time.
2A timing error of ±0.2 s is made on a 2.5 s climb. What is the percentage uncertainty in the power?
0.2 ÷ 2.5 = 0.08, so about ±8% (power is inversely proportional to time).
3Muscles are about 25% efficient. If a student's useful power output is 450 W, estimate her metabolic power during the climb.
Useful = 0.25 × metabolic, so metabolic ≈ 450 ÷ 0.25 = 1800 W. Most becomes thermal energy.
4A student does 30 step-ups onto a 0.25 m step in 40 s. Her mass is 55 kg. Find her power output.
W = 30 × (55)(9.8)(0.25) = 4042.5 J. P = 4042.5 ÷ 40 ≈ 101 W ≈ 1.0 × 10² W.
5A student measured the stairs along the slope (4.5 m) instead of vertically (2.7 m). By what factor was his calculated power too large?
4.5 ÷ 2.7 ≈ 1.7 times too large.
Challenge
1A 70 kg student runs up stairs 3.5 m high in 2.8 s and is moving at 2.0 m/s at the top (starting from rest). (a) Find her power counting only Eg. (b) Find it counting both Eg and Ek. (c) By what percentage did ignoring Ek underestimate the power?
(a) Eg = (70)(9.8)(3.5) = 2401 J; P = 2401 ÷ 2.8 ≈ 858 W. (b) Ek = ½(70)(2.0)² = 140 J; total 2541 J; P ≈ 908 W. (c) (908 − 858) ÷ 908 ≈ 5.5% low.
2A climber gains 900 m of height in 3.0 h on a mountain trail. Her mass plus pack is 80 kg. What is her average useful power? Why is it so much lower than a stair sprint?
W = (80)(9.8)(900) = 705 600 J; Δt = 10 800 s. P ≈ 65 W. Over hours the body can only keep up a much lower rate; short sprints use stored energy quickly and cannot be sustained.
3Design an improvement to the stair investigation that would reduce the timing uncertainty, and explain why it works.
Sample: film the climb at a high frame rate and read the start and finish frames, or use a longer flight of stairs (e.g., two flights). Filming removes reaction-time error; a longer time makes the same ±0.2 s error a smaller percentage of the total.
Quiz
1. To find the work done climbing stairs, h should be:
2. A 60 kg student climbs 3.0 m in 6.0 s. Her power is:
3. Why does the stair-climb calculation underestimate the body's actual energy use?
4. Which change would most reduce the percentage uncertainty from reaction time?
5. Two students do the same work, but A takes half as long as B. Then:
6. A student's power is 600 W. About how many horsepower is that?
Big idea: This is the D2.6 investigation: measuring power output and connecting power to work. It also practises planning, safety, data tables, uncertainty and significant digits (A1.1, A1.4, A1.5, A1.6, A1.8, A1.13).
How to help: Be the timer, and be the safety officer. Insist on dry stairs, a handrail and one step at a time. A slow-walk version gives perfectly good data.
Common difficulties:
- Using the slope length of the stairs instead of the vertical height.
- Using mass instead of weight (multiply by 9.8).
- Treating a heavier student's higher power as "cheating"; they do more work per climb.
- Not recognizing reaction time as a source of error.
Sensitivity: some teens are self-conscious about their mass. They can use a sample mass (for example, 60 kg) instead of their own; the method is what matters.