Adjust mass, drag coefficient, and cross-section area to visualize falling motion in real time. Explore presets from skydiver to raindrop and see how terminal velocity changes.
Presets
Parameters (Custom)
Mass m
Cross-section A
Drag coeff. Cd
Air density ρ (kg/m³)
kg/m³
Results
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Terminal vel. vt (m/s)
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Time to 90% vt (s)
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Re at terminal vel.
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Terminal KE (kJ)
Fall
Velocity v(t)
Acceleration a(t)
Theory & Key Formulas
Equation of motion: $ma = mg - \frac{1}{2}\rho C_d A v^2$
What exactly is terminal velocity? I know things stop accelerating when they fall, but why?
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Basically, it's the speed where the downward pull of gravity is perfectly balanced by the upward push of air resistance. In practice, the object can't go any faster because the drag force gets stronger the faster you go. Try moving the "Mass" slider in the simulator above. You'll see a heavier object reaches a higher terminal velocity because gravity's pull is stronger.
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Wait, really? So the shape and size matter too? How does the "Cross-section A" slider affect things?
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Absolutely. A larger cross-sectional area means the object "rams" into more air molecules every second. For instance, a skydiver spread out like a star creates a huge area and high drag, slowing them down. In the simulator, increase the "Cross-section A" and watch the terminal velocity drop. A common case is a parachute, which dramatically increases area to achieve a safe, low terminal speed.
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What about the "Drag Coefficient" slider? Is that just for how "slippery" or "bluffy" the shape is?
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Exactly! The drag coefficient $C_d$ is a number that captures the shape's aerodynamic efficiency. A sleek, streamlined sports car might have a $C_d$ of 0.25, while a flat plate facing the wind is about 1.28. When you change the $C_d$ parameter from the "Skydiver" to the "Raindrop" preset, you're simulating how a smooth, spherical shape experiences less drag for its size than a tumbling human.
Physical Model & Key Equations
The core physics is Newton's second law applied to a falling object. The net force is weight ($mg$) minus the air drag force, which is proportional to the square of the velocity for high-speed motion. This gives us the equation of motion.
$$ma = mg - \frac{1}{2}\rho C_d A v^2$$
Where $m$ is mass (kg), $a$ is acceleration (m/s²), $g$ is gravity (9.81 m/s²), $\rho$ is air density (kg/m³), $C_d$ is the drag coefficient, $A$ is the cross-sectional area (m²), and $v$ is velocity (m/s).
Terminal velocity ($v_t$) occurs when acceleration is zero. Setting $a=0$ in the equation above and solving for $v$ gives us the terminal velocity formula. This is the key result you can test directly in the simulator.
$$v_t = \sqrt{\dfrac{2mg}{\rho C_d A}}$$
This shows mathematically what we discussed: $v_t$ increases with mass ($m$) and decreases with air density ($\rho$), drag coefficient ($C_d$), and area ($A$).
Frequently Asked Questions
You can simulate the fall of any object by directly editing the mass, drag coefficient, and cross-sectional area input fields. For example, by entering data for a golf ball (mass 0.046 kg, drag coefficient 0.24, cross-sectional area 0.0014 m²), you can check the terminal velocity of an object not included in the presets.
It depends on the object and initial conditions, but for a skydiver (mass 80 kg, drag coefficient 0.7), terminal velocity (approximately 55 m/s) is approached in about 10 to 15 seconds. Use the velocity graph on the simulation screen to gauge the time until the value levels off.
In the current version, the air density is fixed (approximately 1.2 kg/m³ near sea level). At higher altitudes, air density decreases and terminal velocity increases, but this effect is not reflected in this tool. For more accurate high-altitude fall simulations, please use specialized software that allows air density as a variable.
As general guidelines: a sphere (e.g., a baseball) is about 0.47, a skydiver (horizontal posture) is about 0.7 to 1.0, and a streamlined passenger car is around 0.3. A raindrop with a diameter of about 2 mm has a Cd of approximately 0.5. Refer to the simulator's preset values and try numbers close to the actual object.
Real-World Applications
Skydiving & Parachute Design: Skydivers control their terminal velocity by changing their cross-sectional area and drag coefficient. By arching their body, they can reach speeds over 50 m/s. Opening the parachute massively increases $A$ and $C_d$, reducing $v_t$ to a safe ~5 m/s for landing.
Vehicle Aerodynamics: Automotive engineers use wind tunnels and CAE software to minimize a car's drag coefficient ($C_d$) to improve fuel efficiency at high speeds. A lower $C_d$ means the engine works less to overcome air resistance.
Meteorology & Raindrop Physics: A raindrop's terminal velocity (typically 5-9 m/s) determines how fast precipitation reaches the ground. This depends on the drop's size (affecting $m$ and $A$) and shape, which is modeled with a specific $C_d$.
Particle Settling in Industry: In chemical engineering, the terminal velocity concept is used to design separators and settling chambers. For example, calculating how fast dust or powder particles fall through air or liquid is crucial for air pollution control and product processing.
Common Misconceptions and Points to Note
There are a few key points you should be especially mindful of to master this simulator. First, the drag coefficient Cd is not a constant determined solely by shape. For example, while we mentioned a baseball's Cd is about 0.3, this is for a certain velocity range. In reality, it varies with speed, ball spin, and surface roughness. The simulator simplifies this to a constant value, so please treat it strictly as a tool for "grasping trends."
Next, how to interpret the cross-sectional area A. The cross-sectional area here refers to the projected area as seen from the direction of motion. For instance, the air resistance experienced by a skydiver in a spread-eagle position (free-fall) versus a head-first diving position (track position) is completely different. This is because the cross-sectional area A changes dramatically. If you try changing only the area while keeping mass constant in the simulator, you should clearly see its impact on terminal velocity.
Finally, understand the fundamental limitation that reality is far more complex than one-dimensional fall. This calculation uses an idealized model of "falling straight down." For actual baseball breaking balls or lateral movement (drift) in skydiving, forces not handled by this tool—like lift and lateral drag—play a significant role. Please use this as a first step to learn the "basic behavior of air resistance."
Select object preset (skydiver, baseball, raindrop) or enter mass in kg using sMassNum and cross-sectional area in m² using sAreaNum
Adjust drag coefficient (Cd) with sCdNum—typical values: sphere 0.47, skydiver belly-to-earth 1.15, streamlined body 0.04
Set air density (vRho) in kg/m³—use 1.225 for sea level standard atmosphere or 0.4 for 10 km altitude
Simulator computes terminal velocity vt = √(2mg/ρACd), time to reach 90% vt, Reynolds number at terminal velocity, and kinetic energy
Worked Example
A skydiver with mass 80 kg, belly-to-earth area 0.55 m², and Cd 1.15 in standard air (ρ = 1.225 kg/m³): terminal velocity vt = √(2×80×9.81/(1.225×0.55×1.15)) ≈ 53 m/s. Time to 90% vt ≈ 8.2 seconds. At terminal velocity, kinetic energy = 0.5×80×53² ≈ 112 kJ. Reynolds number Re ≈ 1.8×10⁶ (turbulent regime).
Practical Notes
Cd increases with froude number for very high speeds; skydiver in head-down position achieves vt ≈ 90 m/s with Cd ≈ 0.7 and projected area ≈ 0.13 m²
Raindrop (r = 2 mm, mass ≈ 33 mg, Cd ≈ 0.6) reaches vt ≈ 9 m/s; larger drops (r > 6 mm) deform and oscillate rather than reach true equilibrium
Reynolds number at terminal velocity determines flow regime—laminar (Re < 1) for dust particles, transitional (1 < Re < 1000) for small rain, turbulent (Re > 1000) for skydivers and sports balls