Clapping at the Wall: A Speed of Sound That Comes Out Too Low
Two Form 3 students try to measure the speed of sound using the echo from the school boundary wall. They measure the distance to the wall as 85 m, checked twice with a 50 m tape and accurate to about plus or minus 2 m. One student claps and tries to clap again exactly when she hears each echo; the second student starts a stopwatch on the first clap and stops it on the twentieth, reading 13.0 s. They calculate 262 m/s, far below the 340 m/s quoted in their textbook, and conclude that either the tape or the stopwatch must be faulty. Their teacher is unconvinced and asks them to think about what happens between hearing an echo and making the next clap. She also points out that the day is hot (about 30 degrees Celsius) with a steady wind blowing from the wall towards them.
1. Reproduce their calculation and work out how much extra time is being added to each clap-echo cycle.
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Time per cycle: 20 cycles in 13.0 s, so one cycle = 13.0/20 = 0.65 s.
Distance per cycle: the sound travels to the wall and back, so 2 x 85 = 170 m.
Their speed: v = distance/time = 170/0.65 = 262 m/s (about 260 m/s to 2 significant figures).
What it should be: at 340 m/s one cycle should take 170/340 = 0.500 s, and 20 cycles should take 20 x 0.500 = 10.0 s.
Extra time: 13.0 - 10.0 = 3.0 s over 20 cycles = 0.15 s per cycle.
Judgement: 0.15 s is exactly the size of a human reaction time. The measured cycle is not the sound's travel time - it is the sound's travel time plus the time the clapper needs to hear the echo, decide, and move her hands. Every single cycle is stretched by the same amount, in the same direction. This is a systematic error, so the tape and the stopwatch are probably both perfectly good; the method is what is wrong.
2. Explain why averaging over 20 claps does not fix the problem, and assess whether the hot day or the wind could explain the low result.
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Why averaging fails: averaging reduces random error - scatter in reaction time from one clap to the next - but the reaction delay is present in every cycle and always adds time. Averaging 20 stretched cycles just gives a very precise stretched answer. The error is systematic and must be designed out, not averaged out.
The hot day: sound travels faster in warm air, roughly v = 331 + 0.6T m/s, giving about 349 m/s at 30 degrees Celsius. That is higher than 340, so the true cycle time would be 170/349 = 0.487 s, and their answer would be even further below the truth. The temperature cannot explain a low result - it makes the discrepancy larger.
The wind: blowing from the wall towards them, the wind slows the outgoing trip and speeds the return. The two effects nearly cancel for a round trip: the extra time going out is almost exactly recovered coming back, and the remaining effect depends on (wind speed/sound speed)^2, which for a 5 m/s wind is about 0.02% - negligible.
Judgement: neither the weather nor the instruments are the culprit. The students should be directed back to the timing method, and they should stop apologising for their equipment.
3. Propose a revised method that removes the reaction-time error, and say how precise it would be.
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Best fix: remove the human from the loop. Use a smartphone voice recorder placed next to the clapper, clap once sharply, and record the echo. On playback the clap and its echo appear as two spikes, and the gap can be read to about 0.01 s. With the phone's clock measuring the gap, no reaction time is involved at all. At 85 m, v = 170/0.50 = 340 m/s, and a 0.01 s timing error gives only about 2% uncertainty.
Alternative fixes:
- Move much further back. At 250 m one cycle is 0.735 s, so a 0.15 s reaction is a 20% error rather than 30% - better, but still poor.
- Use a visible signal instead of an echo: an assistant drops a raised arm while striking two blocks, and the observer times from seeing the arm to hearing the sound. This removes the echo timing but keeps one reaction delay over a single 0.25 s trip, so it is worse unless the distance is large.
- Have one student clap at a steady rhythm set by a metronome while a second records and later measures the gap between the direct sound and the echo on the recording.
Also improve: measure the distance with a surveyor's tape or GPS rather than twice with a 50 m tape, take several readings and average, and clap once first to check there is a single clean echo rather than several reflections from different buildings.
Judgement: a recorded single clap-and-echo beats every hand-timed version, and it also lets the students see for themselves that the 0.15 s was theirs, not the wall's.