What if a skydiver spread out like a starfish?

A skydiver jumps from a plane. Air rushes past her as she falls.

What if she spreads her arms and legs wide, like a starfish? Will she fall faster, slower or the same?

When skydivers jump from a plane, they fall fast, but not faster and faster forever. The air pushes back on them, and that push gets stronger the faster they go.

A skydiver's body position changes how much of her faces the air. Diving head first, very little does. Spread out wide, like a starfish, much more of her body meets the rushing air. In this lab you can change how much of a skydiver faces the air and see what happens to the top speed.

A falling body in air feels two main forces: its constant weight and an upward drag force that grows with speed. OpenStax describes how the drag on a skydiver increases until it equals the weight; the net force is then zero and the skydiver falls at a constant terminal velocity.

The drag depends on the area facing the air. This lab keeps the skydiver's mass (80 kg) and drag coefficient (1.0) fixed in the fall model and changes only that area, from a compact 0.25 m² to a spread-out 0.7 m², to measure how the terminal speed responds.

First, the skydiver curls up small. Only a little of her body faces the air.

The air does not push back much. So she can fall very fast!

In the model, the skydiver starts curled up, with only a quarter of a square meter of her body facing the air. Her top speed in air, called her terminal speed, works out at about 72 meters per second, around 260 kilometers per hour.

Our lab tower is only 50 meters tall, which is far too short to reach that speed. She lands after about 3.2 seconds at about 30 meters per second, still speeding up. Real skydivers jump from much higher; you can compare the model with real skydiving speeds at the end of this lab.

With m = 80 kg, C_d = 1.0, A = 0.25 m² and ρ = 1.225 kg/m³, the model gives v_t = √(2mg/ρC_dA) = 71.6 m/s, about 258 km/h. On its 50 m drop she lands after 3.24 s at 29.9 m/s, close to the vacuum value of 31.3 m/s, because at these speeds drag is still small compared with her weight.

The model's maximum height of 50 m keeps the run short, so the terminal readout, not the landing speed, shows the speed she would approach on a long fall. Watch the speed graph: it is only just beginning to curve.

Now she spreads out her arms and legs. Much more of her body faces the air.

The air pushes back harder. Her top speed is much lower!

Now spread the skydiver out like a starfish, so 0.7 square meters faces the air, nearly three times as much. Her top speed drops to about 43 meters per second, around 150 kilometers per hour.

On the short 50-meter drop she lands at about 28 meters per second instead of 30, after about 3.3 seconds. The gap grows the longer she falls, because air pushes back harder the faster she goes. Notice that she does not float: she still falls fast. She just stops speeding up sooner.

At A = 0.7 m² the model gives v_t = 42.8 m/s (154 km/h). The ratio 71.6/42.8 = 1.67 equals √(0.7/0.25), since v_t ∝ 1/√A. On the 50 m drop the landing speed falls from 29.9 to 27.6 m/s and the time rises from 3.24 to 3.34 s.

The model can also reproduce OpenStax's worked examples. Set ρ = 1.21 kg/m³: an 85 kg skydiver spread out with C_d = 1.0 and A = 0.70 m² gives 44.4 m/s (OpenStax rounds to 44), and a 75 kg diver head first with C_d = 0.70 and A = 0.18 m² gives 98.2 m/s (OpenStax: 98).

Air pushes on everything that falls. A wide shape catches more air.

When the air pushes up as hard as gravity pulls down, the skydiver stops speeding up.

As the skydiver falls faster, the air pushes back harder. Soon the air's push upward is as strong as gravity's pull downward. The two forces cancel, so she stops speeding up and falls at a steady top speed.

NASA's guide to falling objects explains that something with a large area facing the air falls slower than something small with the same weight. Spreading out gives the air more of her to push on, so the forces balance at a lower speed. Our model is simple: it keeps her shape, the air's thickness and the strength of gravity the same all the way down.

Drag D = ½ρC_dAv² grows with v², while weight mg is constant. NASA Glenn notes that when drag equals weight there is no net force and, by Newton's first law, the object falls at a constant velocity: V = √(2W/C_dρA). A larger A or C_d lowers it; NASA Glenn adds that a large flat plate falls slower than a small ball of the same weight.

NASA Glenn's drag-equation page adds that drag also depends on the air's viscosity and compressibility and on the body's inclination, and that drag coefficients are almost always measured in wind tunnels. Model limits: a constant drag coefficient, constant air density over the fall, a rigid body that does not tumble, and no wind or lift. NASA Glenn also warns that drag rises sharply near the speed of sound, so a drag coefficient measured at low speed gives wrong answers for very high terminal speeds.

Real skydivers spread out to fall more slowly. Diving head first, they fall much faster.

Then they open a parachute. Its big shape catches lots of air, so they float down gently.

OpenStax, a free physics textbook, works out real skydiver speeds. Diving head first, with very little of the body facing the air, a skydiver of 75 kilograms can reach about 350 kilometers per hour. Spread out wide, the top speed may fall to about 200 kilometers per hour.

When the parachute opens, OpenStax explains, the terminal speed becomes much smaller. The parachute's huge area catches so much air that drag balances the skydiver's weight at a gentle speed. Each change of shape is the same trick as our starfish: more area, more drag, a lower top speed.

OpenStax University Physics estimates that a 75 kg skydiver descending head first, with A ≈ 0.18 m² and C ≈ 0.70, reaches about 98 m/s, or 350 km/h, while in a spread-eagle position the terminal velocity may decrease to about 200 km/h. Its worked example puts an 85 kg spread-eagle skydiver (A = 0.70 m², C = 1.0) at 44 m/s. Opening the parachute makes the terminal velocity much smaller.

OpenStax's table of drag coefficients lists a skydiver at about 0.70 feet first and about 1.0 horizontal, so body position changes both C and A. The model holds one fixed shape for the whole fall, so it cannot show a skydiver changing position partway down.