Terminal Velocity — The Complete Physics Guide
Drop an object and, in the idealised vacuum of a physics textbook, it accelerates forever, going faster and faster without limit. In the real world, air resistance changes everything: as an object falls faster, drag force grows, until eventually drag exactly balances gravity and the object stops accelerating altogether, continuing to fall at a constant, maximum speed — its terminal velocity. This single concept explains why a skydiver's fall becomes a steady, controllable descent rather than an ever-accelerating plunge, and why a feather drifts gently down while a stone falls fast and hard.
Terminal velocity depends on a genuinely interesting combination of factors — not just weight, but also shape, size, and the density of the fluid being fallen through — making it a rich, practical application of Newton's second law and fluid drag together.
What is Terminal Velocity?
As a falling object speeds up, the drag force opposing its motion grows roughly with the square of its speed. Eventually, drag force grows large enough to exactly equal the object's weight — at this point, net force is zero, acceleration stops, and the object continues falling at a constant speed: its terminal velocity. This is a direct application of Newton's second law: with zero net force, there's zero acceleration, regardless of how fast the object is actually moving.
Terminal velocity depends on the balance between weight (which depends on mass) and drag (which depends on speed, shape, size, and the density of the surrounding fluid) — a skydiver can noticeably change their terminal velocity just by altering body position, since spreading out (belly-down) increases both drag coefficient and cross-sectional area, reducing terminal velocity, while diving head-first minimises both, increasing it substantially.
The Formula Explained
m is the object's mass. ρ (rho) is the density of the fluid being fallen through — much higher for water than air, which is why terminal velocity in water is dramatically lower than in air for the same object. Cd is the drag coefficient — a dimensionless number describing how streamlined the object's shape is, with lower values indicating less air resistance for the same cross-sectional area. A is the cross-sectional area facing the direction of motion. This formula comes directly from setting weight (mg) equal to drag force (½ρv²CdA) and solving for v.
How to Use This Calculator
Use "m, Cd, A & ρ" to find the terminal velocity of a falling object given its physical properties. Use "v_t, Cd, A & ρ" to find what mass would produce a specific known terminal velocity. Use "v_t, m, Cd & ρ" to find the cross-sectional area needed for a target terminal velocity — useful for parachute and drag-device design.
Worked Example — A Skydiver
Problem: An 80 kg skydiver in belly-down position (Cd ≈ 1.0, A ≈ 0.7 m²) falls through air (ρ = 1.225 kg/m³). Find their terminal velocity.
v_t = √(2mg/(ρCdA)) = √[2(80)(9.81) / ((1.225)(1.0)(0.7))]
v_t = √(1815.5) = 54.0 m/s ≈ 194 km/h — consistent with the well-known typical skydiving terminal velocity
Common Mistakes
Assuming heavier objects always fall faster: in a vacuum, all objects fall at the same rate regardless of mass, but with air resistance, heavier objects of the same shape and size do have higher terminal velocity — the relationship is real but often oversimplified or misunderstood.
Forgetting the square root: terminal velocity is proportional to the square root of mass, not mass directly — doubling mass increases terminal velocity by only about 41%, not 100%.
Real-World Applications
Parachute design: parachutes work by dramatically increasing both drag coefficient and cross-sectional area, drastically reducing terminal velocity to a safe landing speed.
Vehicle aerodynamics: car and aircraft designers minimise drag coefficient to reduce fuel consumption and increase top speed, applying the same physics in reverse — reducing drag rather than maximising it.
Sedimentation and particle settling: geologists and engineers use terminal velocity to predict how quickly particles of different sizes settle out of water or air, relevant to everything from river delta formation to industrial filtration design.
Approaching Terminal Velocity — The Exponential Curve
An object doesn't reach terminal velocity instantly — it approaches it along a smooth curve, accelerating quickly at first (when speed is low and drag is small), then more and more slowly as drag grows and net force shrinks, asymptotically approaching but mathematically never quite reaching the exact terminal value. In practice, an object typically reaches within a few percent of its true terminal velocity after falling a distance that depends on its size and density — a skydiver typically reaches close to terminal velocity within about 12 seconds and 450 metres of freefall, while smaller, less dense objects reach terminal velocity much sooner over a much shorter fall distance.
This approach curve is mathematically similar in shape to other physical processes governed by a balance between a driving force and an opposing resistance that grows with the changing quantity — the same general mathematical pattern appears in charging capacitors, cooling objects, and many other physical systems that settle toward an equilibrium value rather than reaching it in one discrete step.
The Drag Coefficient and Reynolds Number
The drag coefficient (Cd) used in this calculator isn't a perfectly fixed constant for a given shape — it actually depends somewhat on the flow conditions, characterised by a dimensionless quantity called the Reynolds number, which compares inertial forces to viscous forces in the surrounding fluid. At the Reynolds numbers typical of falling skydivers, vehicles, and most everyday objects, drag coefficient is reasonably constant and treating it as fixed (as this calculator does) gives good results. At very low Reynolds numbers (tiny particles settling slowly through a viscous fluid, like dust settling through air, or sediment settling through water) drag behaves quite differently, following Stokes' Law rather than the quadratic drag model used here.
This is why this calculator's formula, while excellent for skydivers, vehicles, and similarly sized falling objects, becomes progressively less accurate for extremely small or extremely slow-moving particles, where a different physical drag regime takes over entirely.
Worked Example 2 — A Raindrop
Problem: A spherical raindrop has mass 6.5×10⁻⁵ kg, drag coefficient 0.47 (typical for a sphere), and cross-sectional area 1.77×10⁻⁵ m². Find its terminal velocity in air.
v_t = √(2mg/(ρCdA)) = √[2(6.5×10⁻⁵)(9.81) / ((1.225)(0.47)(1.77×10⁻⁵))]
v_t ≈ 9.2 m/s ≈ 33 km/h — consistent with the well-documented terminal velocity of medium-sized raindrops, and part of why rain rarely causes injury on impact despite falling from considerable heights
Terminal Velocity at Extreme Altitude — The Red Bull Stratos Jump
In 2012, Felix Baumgartner jumped from a helium balloon at nearly 39 km altitude as part of the Red Bull Stratos project, reaching a peak speed of approximately 1,357 km/h (about Mach 1.25) during his descent — briefly exceeding the speed of sound in air, becoming the first human to do so outside a vehicle. This extraordinary speed was possible precisely because terminal velocity depends on air density: at 39 km altitude, air density is roughly 1,000 times lower than at sea level, meaning drag force was dramatically reduced, allowing Baumgartner to fall far faster than any ordinary skydiver's terminal velocity of around 200 km/h before thickening air at lower altitudes eventually slowed him back down to a normal terminal velocity for his final approach and parachute deployment.
This jump served as a dramatic real-world demonstration of exactly the physics captured in this calculator's formula — as ρ (air density) decreases toward the upper atmosphere, v_t increases correspondingly, since terminal velocity is inversely proportional to the square root of fluid density, with everything else held constant.
Terminal Velocity in Liquids vs Gases
Because terminal velocity depends inversely on the square root of fluid density, and liquids are typically hundreds to thousands of times denser than gases, terminal velocity in a liquid is dramatically lower than in air for the same object — water is roughly 800 times denser than air, so an object's terminal velocity falling through water is only about 1/28th (the square root of 800) of its terminal velocity falling through air, all else being equal. This is why objects appear to fall in almost slow motion underwater compared to through air, and why underwater environments allow far gentler, more controllable descents for divers, submersibles, and sinking debris alike.
This same density-dependence explains why the same drag equation, with appropriate fluid density substituted, applies equally well to objects falling through air, sinking through water, or settling through any other fluid medium — the underlying physics of force balance between weight and drag remains identical regardless of which specific fluid is involved.