Calculator Inputs
Formula Used
Quadratic Drag Model
Quadratic drag is useful for many objects falling through air.
Drag force: Fd = 0.5 × ρ × Cd × A × v²
Terminal velocity: vt = √(2mg / (ρCdA))
Velocity with time: v(t) = vt × tanh(gt / vt)
Time to target fraction: t = (vt / g) × atanh(r)
Distance to target fraction: x = (vt² / g) × ln(1 / √(1 − r²))
Linear Drag Model
Linear drag is useful for slow motion, small particles, and viscous fluid examples.
Drag force: Fd = b × v
Terminal velocity: vt = mg / b
Velocity with time: v(t) = vt × (1 − e−bt/m)
Time to target fraction: t = −(m / b) × ln(1 − r)
Distance to target fraction: x = vt × [t − (m / b) × r]
Here, r is the selected target fraction. For example, 95% means r = 0.95. Time to exactly 100% is infinite in the ideal model.
How to Use This Calculator
- Select quadratic drag for common falling objects in air.
- Select linear drag for slow motion or viscous flow problems.
- Enter mass and gravity in consistent metric units.
- Enter fluid density, drag coefficient, and area for quadratic drag.
- Enter linear drag coefficient when using the linear model.
- Use a target percentage below 100, such as 90, 95, or 99.
- Add a known terminal velocity if you already measured it.
- Press Calculate and review the result above the form.
- Use CSV or PDF export for records, assignments, and reports.
Example Data Table
| Object | Mass kg | Cd | Area m² | Fluid density kg/m³ | Target % | Model |
|---|---|---|---|---|---|---|
| Skydiver spread posture | 80 | 1.0 | 0.70 | 1.225 | 95 | Quadratic |
| Compact ball | 0.145 | 0.47 | 0.0042 | 1.225 | 90 | Quadratic |
| Small viscous particle | 0.002 | Not used | Not used | Not used | 95 | Linear |
Physics Article
Understanding Terminal Velocity
Terminal velocity describes the steady speed reached by a falling body when downward weight equals upward drag. The motion does not stop. The acceleration only falls toward zero as the object approaches a limiting speed. Because the final value is reached asymptotically, a practical calculator should ask for a target percentage, such as 90%, 95%, or 99% of terminal velocity.
Choosing the Drag Model
This calculator supports two common drag models. Linear drag is useful for slow motion, small particles, viscous fluids, or simplified classroom problems. It uses a drag force proportional to velocity. Quadratic drag is better for many real falling objects in air. It uses density, drag coefficient, and cross sectional area. This model is common for balls, skydivers, droplets, and blunt bodies moving through gases.
Important Inputs
Mass has a direct effect on the answer. A heavier object has greater weight, so it often reaches a higher limiting speed. Area and drag coefficient work in the opposite direction. Larger area or larger coefficient creates more resistance. Fluid density also matters. Water produces much stronger drag than air under similar conditions. Gravity changes the result too, so the input can be adjusted for another planet or laboratory setup.
Reading the Result
The time result should be read carefully. Reaching exactly one hundred percent is not finite in the ideal equations. The calculator therefore reports the time needed to reach the selected fraction. A 99% target can take much longer than a 90% target, even when the terminal speed is the same. This helps students see why the curve flattens near the end.
Practical Notes
Distance is included as a useful extra estimate. It shows how far the object may fall before reaching the chosen fraction. This is helpful when checking whether a drop height is enough for terminal velocity to be nearly reached. The result is still an ideal estimate. Real wind, changing density, spin, posture, turbulence, and shape changes can alter the outcome. Use measured data when safety, design, or research decisions require high accuracy.
Better Data Habits
For best results, choose inputs from a consistent source. Estimate area from the widest projected face. Use a drag coefficient that matches the object shape. Then compare several percentages to understand both early acceleration and late approach behavior. Record assumptions with each export.
FAQs
1. Can an object truly reach terminal velocity?
In ideal equations, it approaches terminal velocity but never reaches exactly 100%. That is why this calculator uses a target percentage below 100.
2. What target percentage should I use?
Use 90% for a broad estimate, 95% for common study work, and 99% when you need a closer approach to terminal velocity.
3. Which drag model is more realistic?
Quadratic drag is often more realistic for larger objects moving through air. Linear drag is better for slow motion or viscous flow examples.
4. What does drag coefficient mean?
Drag coefficient describes how shape affects resistance. A streamlined shape has a lower value. A blunt shape usually has a higher value.
5. Why does area matter?
Greater cross-sectional area pushes more fluid aside. That increases drag, lowers terminal velocity, and changes the time needed to approach it.
6. Why is 100% not allowed?
The ideal formulas need infinite time to reach exactly 100%. A percentage below 100 gives a useful finite answer.
7. Can I use a known terminal velocity?
Yes. Enter a measured terminal velocity in the override field. The calculator will use it for the time and distance estimate.
8. Is this calculator safe for engineering design?
Use it for estimates and learning. For safety critical design, confirm results with experiments, standards, or detailed simulation.