How deep is the well?

A stone falls. The sound you hear at the top is produced only at impact. The measured time includes both journeys.

Settings & explanations

Your experiment

Model: released from rest; stationary ideal gas at uniform temperature; no air resistance. Mass changes weight here, but not fall time.

Calculate it yourself

20 °C · 6 s · g = 9.81 m/s²

6 = √(2h / 9.81) + h / c

What is in our atmosphere?

Dry air consists mainly of nitrogen and oxygen. Approximate volume fractions:

N₂O₂

N₂ · 78.08 %   O₂ · 20.95 %   Ar · 0.93 %   CO₂ · approx. 0.04 %

Other trace gases include neon, helium, methane, krypton, hydrogen, nitrous oxide and ozone. The CO₂ fraction changes; 0.04% is a rounded reference value here. Water vapour varies, roughly 0–4% by volume, reducing the percentage shares of the other gases. This applet uses dry air.

Blue: N₂ · Red: O₂ · Purple: Ar · Gold: CO₂ · Turquoise: He. Particle counts, sizes, spacing and motion are schematic. Rare gases are overrepresented for visibility; the percentages above are authoritative. Sound makes particles oscillate locally while the green pressure wave travels.

Sound speed at your temperature

Gasc (m/s)Sound travel time at current depth (s)

Ideal-gas approximation with constant heat capacities near room temperature. CO₂ is a rougher approximation because its heat capacity depends on temperature. Pure gases are comparison models; the atmosphere is a mixture.

Five questions to understand the model

1 · What determines sound speed in gases?

Temperature and gas composition. For an ideal gas, c = √(γ Rₛ T): T in kelvin, Rₛ the specific gas constant and γ the heat-capacity ratio. For dry air we use γ = 1.4 and Rₛ = 287.05 J/(kg·K). At 20 °C, c ≈ 343.23 m/s. Humidity changes composition and therefore c. At fixed temperature and composition, c in an ideal gas is independent of pressure.

2 · How do the stone and sound journeys differ?

Both cover the same distance h in this model, but in opposite directions and one after the other. The stone accelerates: s = ½gt². The sound front moves at constant speed: s = ct. Therefore ttotal = tfall + tsound. Air particles oscillate locally; the pressure disturbance travels upwards.

3 · Does stone mass matter?

Not without air resistance: F = mg = ma → a = g. Mass cancels out. With air resistance, motion depends on mass, shape and frontal area, among other factors.

4 · How do I calculate the well depth?

Substitute h = ½g tfall² and tsound = h/c. This gives tfall = 2ttotal / [1 + √(1 + 2g ttotal/c)]. For 20 °C and 6 s: tfall ≈ 5.55847 s, tsound ≈ 0.44153 s, h ≈ 151.54769 m. Ignoring sound travel time gives 176.58 m — about 25.03 m too much.

5 · What does air resistance do?

A falling stone mainly experiences pressure or form drag and surface friction. A common model is FW = ½ρair cW A v². It includes air density, drag coefficient (shape and flow), frontal area and speed relative to the air. The force opposes motion.

Then approximately m dv/dt = mg − FW. Acceleration is no longer constant; the stone may approach terminal speed. Calculation requires more information and usually a numerical solution. For the same measured total time, ignoring drag overestimates depth when drag slows the fall. This comparison assumes all other conditions are unchanged.

Live values & equations