Drop a feather and a bowling ball from the same height in air, and they land at different times. Aristotle believed heavier objects fall faster by nature. Galileo proved otherwise. The truth involves a beautiful interplay between gravity, air resistance, and a fundamental concept in differential equations: terminal velocity—the speed at which a falling object stops accelerating.
The Forces at Play
When an object falls through air, two forces act on it: gravity pulls it down and air resistance (drag) pushes it up. Gravity is constant: F = mg, where m is mass and g is gravitational acceleration (9.8 m/s² on Earth). Drag is not constant—it increases with velocity. For most objects at everyday speeds, drag follows F_drag = bv, where b is a drag coefficient depending on shape, size, and air density, and v is velocity. The object accelerates as long as gravity exceeds drag, but as speed increases, drag grows until the two forces balance exactly.
The Differential Equation
Newton's second law gives us: m(dv/dt) = mg - bv. This is a first-order linear differential equation. The solution is v(t) = (mg/b)(1 - e^(-bt/m)). This describes velocity starting at 0 and approaching mg/b as t approaches infinity. That limiting value is terminal velocity: v_terminal = mg/b. The exponential decay means the object approaches terminal velocity asymptotically—never quite reaching it mathematically, but effectively achieving it within a few time constants.
Why Mass Matters
Terminal velocity equals mg/b, so heavier objects (larger m) have higher terminal velocities. A bowling ball is denser and heavier than a feather, so it reaches a much higher terminal velocity before drag balances gravity. In a vacuum—where b = 0—both objects accelerate indefinitely at the same rate g, which is why Galileo's vacuum experiment would show equal fall times. On the Moon, astronaut David Scott famously dropped a hammer and feather simultaneously in 1971—they landed together, demonstrating that gravity alone treats all masses equally.
Real-World Applications
Understanding terminal velocity is crucial in many fields. Skydivers reach about 55 m/s (195 km/h) in a spread-eagle position, or 90 m/s (320 km/h) headfirst—and they survive because parachutes dramatically increase the drag coefficient b, lowering terminal velocity to safe landing speeds around 5–6 m/s. Engineers design cars with aerodynamic shapes to minimize b, increasing fuel efficiency at highway speeds. Raindrops fall at terminal velocity—without air resistance, a raindrop falling from a cloud would hit the ground at hundreds of meters per second, turning rain into a lethal event.
Beyond Simple Models
Real drag is more complex than the linear bv approximation. At high speeds, drag scales as bv², giving terminal velocity proportional to sqrt(mg/b) instead of mg/b. Objects can also rotate, creating lift forces (the Magnus effect), explaining why curveballs curve and why a golf ball's dimples extend its range. The full Navier-Stokes equations governing fluid flow around objects are notoriously difficult to solve analytically, which is why aircraft designers rely heavily on computational fluid dynamics simulations.
Conclusion
Terminal velocity elegantly demonstrates how differential equations capture physical reality. The simple equation m(dv/dt) = mg - bv contains the entire story of falling: acceleration from rest, the growing resistance of air, and the asymptotic approach to a final speed where forces balance. It shows how mathematics translates physical intuition—the sense that falling can't accelerate forever—into precise, predictive equations. The feather and bowling ball aren't just a physics demonstration; they're an invitation to see differential equations governing everything that moves through a medium.