Terminal Velocity Calculator
Calculate the terminal velocity of a falling object using mass, projected area, drag coefficient, fluid density, and gravity.
Calculator is for informational purposes only. Terms and Conditions
Choose what to solve for
Select the unknown variable and unit setup before entering known values.
Enter the known values
Only the fields needed for the selected solve mode are shown.
Visual Check
Terminal velocity occurs when upward drag balances downward weight.
Solution
Live result, quick checks, warnings, and full solution steps.
Quick checks
- Check—
Show solution steps See the equation, substitutions, assumptions, and result path
- Enter values to see the full solution steps and checks.
Source, Standards, and Assumptions
Calculation basis, constants, assumptions, and limitations.
Uses the standard educational drag-force relationship \(F_D=\frac{1}{2}\rho C_d A v^2\) and sets drag equal to weight at terminal velocity.
- Assumptions will appear after a valid calculation.
On this page
Calculator Guide
How to Use the Terminal Velocity Calculator
The Terminal Velocity Calculator above estimates the steady falling speed of an object when drag force balances weight. Enter the object mass, projected area, drag coefficient, fluid density, and gravity to calculate terminal velocity, or use a solve mode to work backward for one of those variables.
Use this guide to understand the formula, choose realistic inputs, check units, and decide whether the result is reasonable. The biggest sources of error are usually the drag coefficient \(C_d\), projected area \(A\), and fluid density \(\rho\), so those values deserve the most attention.
Quick Answer
Terminal velocity occurs when the upward drag force equals the downward force of gravity. For quadratic drag, the main formula is \(v_t=\sqrt{\frac{2mg}{\rho A C_d}}\). A larger mass increases terminal velocity, while a larger projected area, higher drag coefficient, or denser fluid lowers it.
For a rough human skydiver estimate, a belly-to-earth falling speed is often around \(50\) to \(55 \, m/s\), or about \(110\) to \(125 \, mph\), but posture and drag assumptions can change the result significantly.
When not to rely on the simplified result
Do not treat terminal velocity as the same as impact speed. A short drop may not give the object enough time or distance to reach terminal velocity. The result is also less reliable if the object tumbles, changes shape, changes posture, reaches very high speed, or moves through fluid density that changes significantly with height.
Inputs and Outputs Used by the Terminal Velocity Calculator
A terminal velocity calculation needs the values that control gravity and drag. The calculator converts selected units internally, but the physical meaning of each input still matters.
| Type | Value | What It Means | Common Units |
|---|---|---|---|
| Input | Mass, \(m\) | The mass of the falling object. Use mass, not weight force. | kg, g, lbm |
| Input | Projected area, \(A\) | The frontal area facing the direction of motion, not the total surface area. | m², ft², cm², in² |
| Input | Drag coefficient, \(C_d\) | A dimensionless value that represents shape, orientation, and flow behavior. | dimensionless |
| Input | Fluid density, \(\rho\) | The density of the air, water, or other fluid surrounding the object. | kg/m³, lb/ft³, slug/ft³ |
| Input | Gravity, \(g\) | The local gravitational acceleration acting downward. | m/s², ft/s² |
| Output | Terminal velocity, \(v_t\) | The estimated steady speed reached when drag balances weight. | m/s, ft/s, mph, km/h |
Terminal Velocity Formula
The terminal velocity formula comes from setting drag force equal to weight. OpenStax presents the quadratic drag relationship as \(F_D=\frac{1}{2}C\rho A v^2\), which is the starting point for this calculator’s main equation.
Main Formula
Use this formula when mass, gravity, fluid density, projected area, and drag coefficient are known.
Force Balance Behind the Formula
At terminal velocity, the net force is zero because the upward drag force equals the downward weight force.
Useful rearranged formulas
The same equation can be rearranged to solve for \(m\), \(A\), \(C_d\), or \(\rho\) when terminal velocity is known. These solve modes are helpful for homework, experiment checks, and comparing object designs.
Source note
For a deeper educational explanation of drag force and terminal speed, see the OpenStax discussion of drag force and terminal speed. For NASA’s explanation of terminal velocity, drag coefficient, reference area, and air density, see the NASA Glenn Research Center terminal velocity reference.
What the Variables Mean
Each variable changes terminal velocity in a predictable direction. Mass and gravity increase the numerator, while fluid density, projected area, and drag coefficient increase the denominator.
\(m\), Mass
Mass measures how much matter is falling. If everything else stays the same, more mass means higher terminal velocity because weight increases.
\(A\), Projected Area
Projected area is the object’s “shadow” facing the flow. For a sphere, use \(A=\pi r^2\), not the full surface area.
\(C_d\), Drag Coefficient
Drag coefficient captures shape and orientation. A streamlined object has a lower \(C_d\), while a flat plate or parachute has a higher one.
\(\rho\), Fluid Density
Fluid density controls how much drag the object experiences. Water is far denser than air, so the same object usually has a much lower terminal velocity in water.
\(g\), Gravity
Gravity converts mass into weight force. Standard Earth gravity is commonly approximated as \(9.80665 \, m/s^2\).
\(v_t\), Terminal Velocity
Terminal velocity is the steady speed where drag and weight balance. It is a limiting speed, not automatically the speed at impact.
How to Use the Calculator
Use the calculator by selecting the unknown, entering realistic known values, checking the units, and reviewing the force-balance quick checks.
Select the solve mode
Choose whether you want to solve for terminal velocity, mass, projected area, drag coefficient, or fluid density. The visible inputs should match the selected unknown.
Enter the object and fluid values
Enter mass, area, \(C_d\), fluid density, and gravity. If the calculator includes object or fluid presets, use them as starting points and adjust values if you have better data.
Check the answer
Confirm that drag force is approximately equal to weight at terminal velocity. Then compare the speed in m/s, mph, and km/h so the answer feels physically realistic.
How to Interpret Terminal Velocity Results
A terminal velocity result tells you the steady falling speed predicted by the simplified drag model. It does not guarantee the object reaches that speed before impact.
What to do with the result
Use it to compare objects, check homework, estimate skydiving-style speeds, or understand how mass, shape, area, and fluid density affect falling motion.
What changes the result most?
Projected area and drag coefficient often dominate because both directly increase drag. Doubling either one lowers terminal velocity by a factor of \(\sqrt{2}\), all else equal.
Sanity check
If a human-sized object in air gives only a few mph or several hundred mph, recheck area, drag coefficient, air density, and unit selections first.
Terminal velocity is not the same as impact velocity
The calculator gives the limiting speed after enough falling distance. For a short drop without much time to accelerate, a Free Fall Calculator can help estimate idealized motion without air resistance, while this calculator shows the drag-limited speed.
Input Checklist Before You Trust the Answer
Terminal velocity is sensitive to input quality. Before using the result, verify that the values describe the actual falling object and fluid.
Use projected area
Use the area facing the flow, not total surface area. For a ball, use the circular cross section. For a person, posture changes the area dramatically.
Use mass, not weight
The formula uses \(m\) as mass. If you have weight force instead, convert carefully or use the force-balance equation directly.
Check the fluid density
Air, water, oil, and other fluids have very different densities. Using air density for water can make the result wildly wrong.
Do not overtrust \(C_d\)
Drag coefficient is not a universal constant. It changes with shape, orientation, surface roughness, Reynolds number, and sometimes speed.
Projected area quick answer
Projected area is the frontal area facing the direction of motion. For a sphere or ball, use \(A=\pi r^2\), not the total surface area \(4\pi r^2\). If you only know diameter, first calculate \(r=\frac{d}{2}\), then calculate area.
Worked Example
This example estimates the terminal velocity of an \(80 \, kg\) person falling belly-to-earth through air. The values are approximate, but the result is in the expected range for a simplified skydiver-style calculation.
Formula
Substitution
Calculation
Final answer
The estimated terminal velocity is \(42.8 \, m/s\), which is about \(95.7 \, mph\). This is reasonable for the selected simplified inputs, although an actual skydiver can be faster or slower depending on posture, clothing, altitude, and drag.
Reverse check
At \(42.8 \, m/s\), the drag force is approximately \(\frac{1}{2}(1.225)(1.0)(0.70)(42.8^2)\approx 785 \, N\). The weight is \(mg=(80)(9.80665)\approx 785 \, N\). Since drag and weight are nearly equal, the result passes the terminal velocity force-balance check.
What the Formula Represents
The terminal velocity equation is a force-balance relationship. Gravity pulls the object downward, drag pushes upward, and the velocity stops increasing when those two forces are equal.
The formula solves the speed where \(F_D=mg\). Above that speed drag would exceed weight; below that speed weight is still larger than drag.
Reference Checks and Typical Values
Reference values are useful for spotting bad inputs, but they are not universal constants. Shape, posture, altitude, surface roughness, and fluid conditions all matter.
| Situation | Typical Check | Why It Matters |
|---|---|---|
| Air near sea level | \(\rho\) is often near \(1.2 \, kg/m^3\) | Air density directly affects drag and changes with altitude, temperature, and pressure. |
| Sphere or ball | Use projected area \(A=\pi r^2\) | The frontal circular area controls drag, not the full sphere surface area. |
| Human belly-to-earth | Often roughly in the 50 m/s range | Actual speed varies with posture, clothing, mass, and altitude. |
| Parachute | Large \(A\) and high \(C_d\) | Parachutes reduce terminal velocity by increasing drag area and drag coefficient. |
Sphere drag coefficient
A simplified sphere value is often near \(C_d=0.47\), but the exact value depends on Reynolds number, surface condition, and flow behavior.
High-drag shapes
Flat plates, broad body positions, and parachutes can use much larger effective drag values, but the correct \(C_d\) depends on the reference area used.
Practical note: NASA’s terminal velocity material explains that increasing reference area and drag coefficient lowers terminal velocity. This is why parachutes work: they create much more drag for the same falling mass.
Design Notes and Practical Ranges
For most users, this page is an educational and estimation tool, not a final aerodynamic design model. The simplified equation is useful when the object has a reasonably steady orientation and the drag coefficient can be treated as approximately constant.
Good use cases
Homework checks, skydiving-style estimates, ball or sphere comparisons, parachute demonstrations, and quick sensitivity studies.
Use more detailed analysis when
The object is tumbling, compressibility matters, the fluid density changes significantly, or the drag coefficient is not stable.
Units and Conversions
The cleanest manual calculation uses SI units: kg for mass, m² for projected area, kg/m³ for density, m/s² for gravity, and m/s for terminal velocity.
Common unit traps
Do not mix lbm with lbf, ft² with in², or mph with m/s inside the same substitution. If you calculate by hand, convert everything to SI first, solve for \(v_t\) in m/s, then convert the final result to mph, ft/s, or km/h.
Speed conversions
\(1 \, m/s = 2.23694 \, mph\), \(1 \, m/s = 3.28084 \, ft/s\), and \(1 \, m/s = 3.6 \, km/h\).
Area conversions
\(1 \, ft^2 = 0.092903 \, m^2\) and \(1 \, in^2 = 0.00064516 \, m^2\). Area unit mistakes can strongly distort drag.
Terminal Velocity vs Free Fall, Drag Force, and Impact Energy
Terminal velocity is related to other motion calculations, but it answers a specific question: the drag-limited steady speed.
Free fall
Ideal free fall ignores air resistance. It is useful for short drops or classroom motion checks, but it does not include drag.
Terminal velocity
Terminal velocity includes drag and finds the speed where drag equals weight.
Kinetic energy
After finding speed, use a Kinetic Energy Calculator to estimate motion energy from mass and velocity.
Common Mistakes
Most terminal velocity mistakes come from using the wrong area, guessing drag coefficient too casually, or mixing unit systems.
Do
- Use projected area facing the flow.
- Convert all hand-calculation inputs to SI units.
- Use realistic \(C_d\) values for the object’s shape and orientation.
- Check whether drag force approximately equals weight at the result.
Don’t
- Do not use total surface area when the formula needs projected area.
- Do not enter diameter where the calculator asks for projected area. For a circular object, calculate \(A=\pi r^2\) first.
- Do not use weight in pounds-force as if it were mass in pounds-mass.
- Do not assume the object reaches terminal velocity from a short drop.
- Do not treat \(C_d\) as exact for tumbling or changing orientations.
Troubleshooting Unrealistic Results
If the terminal velocity looks too high, too low, or physically suspicious, check the variables that most strongly control drag: projected area, drag coefficient, and fluid density.
Result is too high
Check whether projected area is too small, drag coefficient is too low, fluid density is too low, or mph was accidentally used as m/s.
Result is too low
Check whether area is too large, the object was modeled like a parachute, density was set to water instead of air, or mass was entered too small.
Result changes drastically
That may be real. Because \(v_t\) uses a square root, a four-times change in area, density, or \(C_d\) creates about a two-times change in terminal velocity.
Result is blank
Check for zero, negative, missing, or disabled inputs. Terminal velocity requires positive mass, area, drag coefficient, density, and gravity.
Result is near or above Mach 1
The simplified constant-\(C_d\) incompressible model may no longer be appropriate. A Mach Number Calculator can help compare speed with the local speed of sound.
Assumptions and Limitations
This calculator uses a simplified quadratic drag model. It is best for educational estimates and early engineering checks, not final aerodynamic design or safety-critical analysis.
Constant drag coefficient
The model assumes \(C_d\) stays constant. In reality, it can change with speed, Reynolds number, surface roughness, and orientation.
Constant fluid density
The model assumes one density value. Air density changes with altitude, temperature, humidity, and pressure.
Steady orientation
The model assumes the projected area and drag behavior are stable. Tumbling objects can have changing drag and changing area.
Enough fall distance
The calculator gives the limiting speed, not the actual impact speed from a specific drop height.
Not final design approval
For parachute systems, aerospace design, safety equipment, or field-critical applications, use appropriate testing, standards, manufacturer data, and professional review.
Key Terms
These terms help connect the calculator inputs, formula, and result.
Terminal velocity
The steady falling speed where drag force equals weight and acceleration becomes zero.
Drag force
The resistive force from a fluid acting opposite the object’s motion.
Projected area
The frontal area facing the flow, often described as the object’s shadow in the direction of motion.
Drag coefficient
A dimensionless value that represents how shape and orientation affect drag.
Fluid density
The mass per unit volume of the surrounding air, water, or other fluid.
Quadratic drag
A drag model where drag force is proportional to velocity squared.
FAQ
What is terminal velocity?
Terminal velocity is the steady falling speed an object reaches when upward drag force equals downward weight. At that point, net force is zero and the object stops accelerating.
What formula does the Terminal Velocity Calculator use?
For quadratic drag, the calculator uses \(v_t=\sqrt{\frac{2mg}{\rho A C_d}}\), where \(m\) is mass, \(g\) is gravity, \(\rho\) is fluid density, \(A\) is projected area, and \(C_d\) is drag coefficient.
What is the terminal velocity of a human?
A common rough estimate for a belly-to-earth skydiver is about \(50\) to \(55 \, m/s\), or about \(110\) to \(125 \, mph\), but the value changes with mass, posture, clothing, altitude, projected area, and drag coefficient.
Does mass affect terminal velocity?
Yes. If drag coefficient, projected area, fluid density, and gravity stay the same, increasing mass increases terminal velocity because a larger weight force requires more drag to balance it.
Does every falling object reach terminal velocity?
No. Terminal velocity is the limiting steady speed. An object dropped from a short height may hit the ground before it reaches that speed.