Free Fall with Air Resistance Calculator
See how air drag changes a fall: speed, distance, time to approach terminal velocity, and how it compares with a fall in a vacuum.
🪂 What is Free Fall with Air Resistance?
Free fall with air resistance describes an object that falls under gravity while the air pushes back on it. In a vacuum the speed would keep growing as g times t forever. In air, the drag force grows with the square of the speed, so the net downward force shrinks as the object speeds up. The acceleration starts at g, falls gradually, and tends to zero as the speed approaches the terminal velocity. This calculator solves that motion exactly for quadratic drag, the regime that applies to people, balls, parcels and most everyday objects falling at speeds above a few metres per second.
The exact solution uses the hyperbolic tangent. The speed after a time t is v_t tanh(g t / v_t), and the distance fallen is (v_t squared / g) times the natural log of cosh(g t / v_t). You can also invert it to find the time needed to fall a given height. This page offers both directions: enter a time and read the speed and distance, or enter a height and read the fall time and impact speed. Every result is shown next to the vacuum value so you can see directly how much the air changes the outcome, and two charts compare the drag fall with the vacuum fall.
It is useful for physics coursework, for estimating how long a package or a person takes to fall a given distance, for comparing the fall of different objects, and for judging whether the simple vacuum formulas are good enough. If your fall is short, much shorter than the time to reach 95 percent of terminal speed, the vacuum model is already accurate and the free fall calculator is enough. If you only want the limiting speed and how it scales with mass and area, use the terminal velocity calculator, which this page deliberately does not duplicate.
A few caveats matter. The model keeps the drag coefficient and the air density constant, so a very long fall through thinning air is only approximate. It ignores buoyancy, wind and any change of shape during the fall, and above roughly 100 m/s in air the drag coefficient itself begins to change. For tiny particles in viscous fluids, where drag is proportional to speed, a different law applies. For everything else, the results here are a physically sound estimate.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Skydiver Ten Seconds After Release
80 kg, Cd = 1.0, A = 0.7 m², sea level air, t = 10 s
Example 2 — Baseball Dropped for Three Seconds
m = 0.145 kg, Cd = 0.4, A = 0.0042 m² (assumed values), sea level air, t = 3 s
Example 3 — Skydiver Falling 300 Metres
80 kg, Cd = 1.0, A = 0.7 m², sea level air, h = 300 m
Example 4 — A Ten Metre Drop Is Almost a Vacuum Fall
80 kg, Cd = 1.0, A = 0.7 m², sea level air, h = 10 m
Example 5 — Baseball Falling 50 Metres
m = 0.145 kg, Cd = 0.4, A = 0.0042 m², sea level air, h = 50 m
❓ Frequently Asked Questions
🔗 Related Calculators
What is the equation for free fall with air resistance?
With quadratic drag the speed is v(t) = v_t tanh(g t / v_t) and the distance fallen is y(t) = (v_t squared / g) ln cosh(g t / v_t), where v_t = sqrt(2 m g / (rho Cd A)). It solves m dv/dt = m g minus half rho Cd A v squared.
How do you calculate speed after a given time with drag?
Find the terminal velocity v_t first, then use v = v_t tanh(g t / v_t). For 80 kg, Cd 1.0, 0.7 m2 at sea level, v_t is 42.78 m/s and after 10 s the speed is 41.91 m/s, compared with 98.07 m/s in a vacuum.
How long does it take to reach terminal velocity?
Strictly never, but 95 percent of v_t is reached at t = (v_t / g) atanh(0.95), about 1.83 v_t/g. For a 42.78 m/s skydiver that is 7.99 s, during which the fall covers about 217 m.
How far do you fall in a given time with air resistance?
Use y = (v_t squared / g) ln cosh(g t / v_t). The skydiver above falls 300.3 m in 10 s, compared with 490.3 m in a vacuum. Once at terminal speed, the distance grows by v_t metres every second.
How long does a fall from a height take with drag?
Use t = (v_t / g) acosh(exp(g h / v_t squared)). The same skydiver needs 9.99 s to fall 300 m, instead of 7.82 s in a vacuum, and lands at 41.91 m/s instead of 76.71 m/s.
Does air resistance matter for short drops?
Only a little for dense objects. A skydiver dropped 10 m lands at 13.64 m/s with drag instead of 14.00 m/s in a vacuum, and the fall takes 1.44 s instead of 1.43 s. The effect grows with speed because drag grows with speed squared.
Why does the speed follow a tanh curve?
Drag grows as v squared, so the acceleration is g(1 minus v squared over v_t squared). Integrating that equation gives the hyperbolic tangent, which rises almost linearly at first and flattens smoothly toward the terminal speed.
How is this different from the terminal velocity calculator?
The terminal velocity calculator focuses on the limiting speed, unit conversions and how it changes with mass or area. This one models the whole fall, adds the height-to-time mode, the acceleration and drag force along the way, and a direct vacuum comparison.
What happens to the acceleration as the object falls?
The net acceleration is a = g (1 minus (v / v_t) squared) = g / cosh squared(g t / v_t). It starts at g, falls to about half of g when v is 71 percent of v_t, and approaches zero at terminal speed.
Can I use this for a ball or other small object?
Yes, as long as the flow is in the quadratic regime (Reynolds number above a few thousand) and the speed is below about 100 m/s. A baseball with m 0.145 kg, Cd 0.4 and A 0.0042 m2 has a terminal speed near 37 m/s in sea level air. These Cd and area values are assumptions.
What are the limits of this model?
It assumes a constant drag coefficient, constant air density over the fall, no buoyancy and no wind. For very long falls the air density changes, and above about 100 m/s in air compressibility changes Cd. Treat those results as estimates.