• Terminara – Terminal Velocity Physics Sticker

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Terminara

Terminal Velocity Physics

\[m\frac{dv}{dt} = mg - \tfrac12\rho C_d A\,v^2\]
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Why does a skydiver stop speeding up? Air drag grows with the square of the speed. When drag balances weight, the net force is zero and the fall settles at terminal velocity, v_t = √(2mg/ρC_dA). For an 80 kg skydiver falling belly-down in sea-level air, that is about 53 m/s, roughly 190 km/h. A 3" pixel-art sticker; equation, symbols and sources in the physics panel below.

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Size3″ × 3″

3″ × 3″ kiss-cut sticker.

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The physics

A falling object speeds up until air drag, which grows with the square of speed, balances its weight. From then on it falls at a steady terminal velocity, approaching it along a tanh curve.

Terminara has stopped speeding up. She is still falling, just not any faster.

Air pushes back on a falling body with a drag force ½ρC_dA v² that grows with the square of the speed. When drag equals weight, the net force is zero and the speed stops changing. That speed is the terminal velocity, v_t = √(2mg/ρC_dA).

With the illustrative values in the art, an 80 kg skydiver lying belly-down (C_dA = 0.45 m²) in sea-level air, v_t is 53.4 m/s, about 192 km/h. Falling from rest the speed follows v = v_t tanh(gt/v_t) and reaches 99 % of v_t after about 14 s. The graph on her sticker is that curve, checked against a numerical solution.

Real skydivers change v_t by changing shape: tucking in shrinks C_dA and they fall faster.

I stopped speeding up.

Equations

\[m\frac{dv}{dt} = mg - \tfrac12\rho C_d A\,v^2\]
\[v_t = \sqrt{\frac{2mg}{\rho C_d A}}\]
\[v(t) = v_t\tanh\!\left(\frac{g t}{v_t}\right)\]
SymbolMeaningUnit
\(m\) mass of the skydiver (here 80 kg) \(\mathrm{kg}\)
\(v\) falling speed \(\mathrm{m/s}\)
\(t\) time since leaving the plane, falling from rest \(\mathrm{s}\)
\(g\) gravitational acceleration \(\mathrm{m/s^2}\)
\(\rho\) air density (here 1.225 kg/m^3, sea-level standard) \(\mathrm{kg/m^3}\)
\(C_d A\) drag coefficient times frontal area (here 0.45 m^2, belly-down) \(\mathrm{m^2}\)
\(v_t\) terminal velocity (here 53.4 m/s, about 192 km/h) \(\mathrm{m/s}\)

Sources

  1. NASA Glenn Research Center, Beginner’s Guide to Aeronautics: Terminal Velocity
  2. Taylor, Classical Mechanics (2005), Sec. 2.4: Quadratic air resistance

Source Newton, I. (1687). Philosophiæ Naturalis Principia Mathematica, Book II, Section II (motion of bodies resisted in the duplicate ratio of their velocities). London: Royal Society (J. Streater).

Field notes

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