Free Fall, Air Resistance, and Terminal Velocity — The Complete Physics Guide
Free Fall Race is a prediction game — you make your call about which object lands first, then watch the real physics play out frame by frame. Every object's motion is governed by Newton's second law with a genuine drag force, F_d = ½ρC_dAv², recalculated continuously as each object falls. It's built to correct one of the most persistent physics misconceptions people carry from everyday experience: the belief that heavier objects simply fall faster.
Free Fall in a Vacuum
In the absence of air resistance, every object accelerates downward at exactly the same rate, g = 9.81 m/s², regardless of its mass. This is a direct consequence of the equivalence principle: gravitational force is proportional to mass (F = mg), and by Newton's second law, acceleration is force divided by mass (a = F/m) — the mass cancels out completely, leaving a = g for every object, from a feather to a cannonball.
From rest, the standard SUVAT kinematics apply directly: v = gt, s = ½gt², and v² = 2gs. If an object starts with some initial velocity, the same equations apply with that initial velocity included — free fall is just constant-acceleration motion with a = g.
This isn't just theoretical. Apollo 15 astronaut David Scott demonstrated it live on the Moon in 1971, dropping a geological hammer and a falcon feather from the same height at the same instant — with no atmosphere to interfere, they landed together, exactly as Galileo had argued more than three centuries earlier using inclined planes and a much subtler experimental approach.
Air Resistance and Drag Force
Real falls happen in air, and air pushes back. The drag force on a falling object is F_d = ½ρC_dAv², where ρ is air density (about 1.2 kg/m³ at sea level), C_d is a dimensionless drag coefficient that depends on shape (a sphere is roughly 0.47, a streamlined teardrop closer to 0.04), A is the object's cross-sectional area facing the airflow, and v is its current speed.
The critical detail is that drag scales with the square of velocity — double your falling speed and drag quadruples. This means the net downward acceleration, a = g − F_d/m, isn't constant like it is in a vacuum: it starts at close to g when speed is low and drag is negligible, then steadily shrinks as speed (and therefore drag) builds up.
This is exactly the calculation this game runs every single frame for every falling object: current speed determines current drag, which determines current net force, which determines the next instant's acceleration. It's a genuine numerical simulation, not a lookup table or a canned animation — which is why objects with different shapes and masses race to the ground in ways that actually match real-world physics.
Terminal Velocity
As a falling object speeds up, drag grows until it exactly equals the object's weight — at that point, net force is zero, acceleration stops, and the object continues falling at a constant speed called terminal velocity: v_t = √(2mg/(ρC_dA)). Notice that terminal velocity increases with the square root of mass and decreases with the square root of area — heavier or more streamlined objects reach a higher terminal velocity than lighter or more spread-out ones.
This single formula explains a huge range of everyday observations. A skydiver in a standard belly-down, spread-eagle position reaches roughly 55 m/s (200 km/h); the same skydiver falling head-down, presenting a much smaller cross-sectional area, can reach 90 m/s or more. Opening a parachute increases effective area from roughly 0.7 m² to 50 m² or more, dropping terminal velocity to a survivable 4–6 m/s almost immediately.
Raindrops offer a particularly striking example: without air resistance, a raindrop falling from a 2 km cloud base would strike the ground at roughly 198 m/s — comparable to a rifle bullet, and lethal on contact. Air resistance limits real raindrops to a terminal velocity of just 2–9 m/s depending on droplet size, which is precisely why rain is merely uncomfortable rather than deadly.
Stokes' Drag — A Different Regime
The ½ρC_dAv² formula describes drag at the speeds and sizes typical of everyday falling objects, but it isn't universal. For very small objects moving slowly through a fluid — a dust particle settling through air, a fog droplet, or a cell moving through a growth medium — drag instead follows Stokes' law: F_d = 6πηrv, proportional to velocity itself rather than velocity squared, where η is the fluid's viscosity and r is the object's radius.
Which regime applies depends on the Reynolds number, a dimensionless ratio of inertial to viscous forces. At low Reynolds numbers (small, slow objects in a viscous fluid), Stokes' linear drag dominates; at high Reynolds numbers (the falling objects in this game), the quadratic ½ρC_dAv² relationship takes over. Both are genuinely "air resistance" — they just describe different physical regimes of the same underlying fluid dynamics.
Worked Example — Terminal Velocity of a Skydiver
Problem: A skydiver of mass 80 kg falls belly-down with cross-sectional area A ≈ 0.7 m² and drag coefficient C_d ≈ 1.0. Taking air density ρ = 1.2 kg/m³, find the terminal velocity.
v_t = √(2mg / (ρC_dA)) = √((2 × 80 × 9.81) / (1.2 × 1.0 × 0.7))
v_t = √(1569.6 / 0.84) = √1868.6 ≈ 43.2 m/s (about 155 km/h) — close to the real-world figure for a stable belly-down freefall position.
Real-World Applications
Vehicle aerodynamics: Car and aircraft designers spend enormous effort minimising drag coefficient C_d, because at highway and cruising speeds, drag force (and the fuel needed to overcome it) grows with the square of speed — a small reduction in C_d translates into a meaningful, permanent fuel saving across the vehicle's entire operating life.
Parachute and drogue design: Every parachute is engineered around this exact terminal-velocity formula, sized to bring a payload — whether a skydiver, a returning spacecraft capsule, or an air-dropped supply crate — down to a survivable landing speed by maximising effective area A.
Sports ball design: The dimples on a golf ball are a deliberate drag-management trick — they create a thin turbulent boundary layer that actually reduces overall drag compared to a smooth sphere at typical golf-ball speeds, letting a well-struck ball fly significantly farther than an equivalent smooth ball would.
Felix Baumgartner's 2012 stratospheric jump: Jumping from 39 km altitude, where air density is only a tiny fraction of sea-level values, Baumgartner briefly exceeded the speed of sound in free fall — something impossible lower in the atmosphere, where dense air produces enough drag to cap terminal velocity well below Mach 1. As he fell into thicker air, drag rapidly increased and his speed fell back toward a normal terminal velocity, exactly as the F_d = ½ρC_dAv² relationship predicts for increasing ρ.