Table of Contents
Free Fall with Drag
Real free fall includes air resistance (drag force), which increases with the square of velocity. As an object accelerates, drag increases until it equals gravitational force. At this point, acceleration stops and the object falls at constant terminal velocity. This is why skydivers reach about 55 m/s rather than accelerating indefinitely.
The drag force is F_d = 1/2 * rho * Cd * A * v^2, where rho is air density, Cd is the drag coefficient (shape-dependent), A is cross-sectional area, and v is velocity. Terminal velocity occurs when mg = F_d, giving vt = sqrt(2mg / (rho*Cd*A)).
Formulas
Terminal Velocities
| Object | Terminal Velocity |
|---|---|
| Raindrop (2mm) | 6.5 m/s |
| Skydiver (belly) | 55 m/s |
| Skydiver (head-down) | 90 m/s |
| Baseball | 42 m/s |
| Golf ball | 32 m/s |
FAQ
How long to reach terminal velocity?
An object reaches 95% of terminal velocity after falling for a time t = atanh(0.95)/sqrt(g*k) where k = rho*Cd*A/(2m). For a skydiver, this takes about 12-15 seconds and roughly 450 meters of fall.
Does shape matter?
Enormously. Drag coefficient Cd ranges from 0.04 (streamlined body) to 2.1 (flat plate). A sphere has Cd = 0.47. A skydiver in belly position has Cd ~1.0; tucked position Cd ~0.4. Parachutes have Cd ~1.5-2.0 with large area, dramatically reducing terminal velocity.
How does altitude affect terminal velocity?
Air density decreases with altitude. At 10 km altitude, density is about 40% of sea level, increasing terminal velocity by about 60%. This is why Baumgartner reached supersonic speed falling from 39 km where air density is less than 1% of sea level.