How to use the terminal velocity calculator
- Enter the mass of the falling object in kilograms, the frontal area it presents to the air in square meters and the drag coefficient, a shape factor with no unit.
- Enter the air density. The example value is for air near sea level on a mild day, and it is lower at altitude.
- Read the terminal speed, the speed at which the drag force equals the weight and the object stops speeding up.
- Use the table to see how long and how far the object falls before it reaches 50, 90, 95 and 99 percent of that speed, and open the advanced option to see the speed and distance at a moment you choose.
Formula
terminal speed v_t = sqrt(2 x m x g / (rho x Cd x A))
speed after t seconds v(t) = v_t x tanh(g x t / v_t)
distance fallen s(t) = (v_t^2 / g) x ln(cosh(g x t / v_t))
time to reach a share p of v_t: t = (v_t / g) x atanh(p)
m is mass, g is 9.80665 m/s2, rho is the air density, Cd is the drag coefficient and A is the frontal area. The model has drag that grows with the square of speed and stays the same shape throughout the fall.
An 80 kg skydiver, belly down
Take a mass of 80 kg, a frontal area of 0.7 m2, a drag coefficient of 1.0 and an air density of 1.225 kg/m3. The terminal speed is the square root of 2 x 80 x 9.80665 / (1.225 x 1.0 x 0.7), which is 42.78 m/s, or 154.0 km/h, or 95.7 mph. After 10 seconds the speed is 41.91 m/s and the skydiver has fallen 300.3 m. The 99 percent point arrives after 11.54 seconds and about 365 m.
Pulling in the arms and legs changes the numbers a great deal. With a frontal area of 0.18 m2 and a drag coefficient of 0.7, the terminal speed rises to 100.8 m/s, about 363 km/h. A baseball of 0.145 kg with an area of 0.00426 m2 and a coefficient of 0.35 tops out near 39.5 m/s, about 88 mph. All of these values are examples you can replace.
Why heavier objects fall faster in air
In a vacuum every object falls at the same rate. In air, drag pushes back on a falling object more and more as it speeds up, until the push equals the weight. The balance point depends on the weight and on how much air the object has to push aside. Doubling the mass at the same shape raises the terminal speed by the square root of 2, a factor of about 1.41.
Thin air at altitude has a lower density, so the same body falls faster. At 0.7 kg/m3 the skydiver in the example would reach about 56.6 m/s. This is why the early part of a high jump is faster than the same fall near the ground.
Limits of this model
Real objects change their drag coefficient with speed and with tumbling, and the density of the air changes during a long fall. A person can also change shape on purpose. The tool treats all of these as fixed. It is meant for study, homework and curiosity, and it must never be used to plan a jump, a drop test or any activity where safety depends on the answer.
More motion and materials tools
For other forces on moving objects and materials, try the Buoyancy and Hydrostatic Pressure Calculator, Spring Oscillation Calculator and Thermal Expansion Calculator.
Assumptions and limits
- The object falls straight down from rest, with a constant frontal area and a constant drag coefficient.
- Drag is proportional to the square of the speed, which suits fast motion of a bulky object through air.
- The air density is the same at every height of the fall.
- The acceleration of gravity is 9.80665 m/s2 at every height.
- The mass, area, drag coefficient and density in the form are examples, not measured values for any real object.
Frequently asked questions
What is a drag coefficient?
It is a number with no unit that describes how streamlined a shape is. A smooth teardrop has a small value and a flat plate facing the wind has a large one. Measured values for real shapes depend on the speed and the surface, so use a figure from a source you trust.
Does an object ever exactly reach terminal speed?
In this model it only approaches it. After a few multiples of the time shown in the 99 percent row, the difference is too small to notice. That is why the table stops at 99 percent and does not claim 100.
Why does the page ask for frontal area and not size?
Drag depends on how much air the object meets head on. A long thin rod falling nose first has a small frontal area, and the same rod falling flat has a large one.
Can I use this for a raindrop or a feather?
You can enter the numbers, but small light things reach low speeds where drag no longer grows with the square of speed. The result will be a rough guide at best for those cases.