Terminal Velocity
Terminal velocity occurs when drag force equals weight: mg = ½ρv²AC_d. Solve for v = √(2mg/(ρAC_d)). Skydiver ~55 m/s, raindrop ~9 m/s.
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v_term = √(2mg/(ρAC_d)) Higher C_d or area → lower terminal velocity Altitude reduces ρ, increases v_term Time to reach ~99% of v_term is ~5τ
Ready to run the numbers?
Why: Terminal velocity determines impact speed for skydivers, raindrops, and falling objects. Essential for safety and design.
How: Set weight = drag: mg = ½ρv²AC_d. Solve for v. Drag coefficient C_d depends on shape (sphere ~0.47, skydiver ~1.0).
Run the calculator when you are ready.
🎯 Quick Object Selection
⚙️ Object & Environment Parameters
📊 Terminal Velocity Results
📈 Velocity Analysis
🚀 Velocity vs Time
Approaching terminal velocity asymptotically
📏 Distance vs Time
⚖️ Force Balance
At terminal velocity: Weight = Drag
🔄 Comparison
📝 Step-by-Step Solution
Mass (m): 80.0000 kg
Drag Coefficient (Cd): 1.000
Cross-sectional Area (A): 0.700000 m²
Air Density (ρ): 1.2250 kg/m³
Gravity (g): 9.81 m/s²
At terminal velocity, drag force equals weight:
F_drag = F_weight → ½\text{rho} v^{2} ext{CdA} = ext{mg}
Solving for terminal velocity:
v_t = √(2mg / \text{rho} ext{CdA})
Substituting values:
v_t = √(2 × 80.00 × 9.81 / (1.2250 × 1.00 × 0.700000))
→ v_t = 42.7836 m/s
Converting to other units:
→ 154.0 km/h = 95.7 mph
Weight: W = mg
→ W = 784.8000 N
Drag force at terminal velocity: F_d = ½ρv²CdA
→ F_d = 784.8000 N
Verification: W ≈ F_d
→ 784.8000 ≈ 784.8000 ✓
Characteristic time constant: τ = m / (ρCdAv_t)
→ τ = 2.18 s
Time to reach ~99% terminal velocity: t ≈ 5τ
→ t ≈ 10.90 seconds
Approximate distance to terminal velocity:
→ d ≈ 233.2 meters
📖 What is Terminal Velocity?
Terminal velocity is the constant speed that a freely falling object reaches when the resistance of the medium (usually air) prevents further acceleration. It's the maximum velocity an object achieves during free fall, where the upward drag force equals the downward gravitational force.
🎯 Key Concepts
- • Drag force increases with the square of velocity
- • Terminal velocity is reached when drag force = weight
- • Depends on mass, shape, size, and air density
- • Higher at higher altitudes due to thinner air
- • Objects with larger area-to-mass ratios fall slower
- • Streamlined objects have higher terminal velocities
🪂 Skydiving Facts
- • Belly-to-earth position: ~55 m/s (200 km/h)
- • Head-down dive position: ~90 m/s (320 km/h)
- • Felix Baumgartner reached 373 m/s (1,342 km/h)!
- • Takes approximately 10-15 seconds to reach terminal
- • Body position dramatically changes terminal velocity
- • Wingsuits can achieve glide ratios of 3:1
The concept of terminal velocity is crucial in many applications, from skydiving safety to understanding raindrop sizes, designing parachutes, and even analyzing the fall behavior of animals.
🌧️ Raindrops
Small raindrops (1mm) reach ~4 m/s, while large ones (5mm) reach ~9 m/s. Without terminal velocity, raindrops would hit the ground at lethal speeds!
🐱 Cats
Cats have a relatively low terminal velocity (~27 m/s) due to their ability to spread their bodies. This is why cats can survive falls from significant heights.
🦅 Falcons
The peregrine falcon is the fastest animal, diving at over 390 km/h by tucking its wings to minimize drag and maximize terminal velocity.
🧮 How Terminal Velocity Works
When an object falls through a fluid (like air), two main forces act on it: gravity pulling it down and drag pushing it up. As velocity increases, drag increases until the forces balance.
Terminal Velocity Formula
Derivation Steps
- Weight force: F_w = mg (downward)
- Drag force: F_d = ½ρv²CdA (upward)
- At terminal velocity: F_w = F_d
- mg = ½ρv_t²CdA
- Solve for v_t: v_t = √(2mg/ρCdA)
Key Relationships
- • ↑ Mass → ↑ Terminal velocity
- • ↑ Drag coefficient → ↓ Terminal velocity
- • ↑ Cross-sectional area → ↓ Terminal velocity
- • ↑ Air density → ↓ Terminal velocity
- • ↑ Altitude → ↓ Air density → ↑ Terminal velocity
Velocity as Function of Time
Velocity over time:
v(t) = v_t × tanh(gt / v_t)
Distance over time:
d(t) = (v_t²/g) × ln(cosh(gt/v_t))
⏰ When to Use This Calculator
🎓 Education
- • Physics homework and labs
- • Understanding drag forces
- • Comparing free fall vs. with air resistance
- • Fluid dynamics concepts
- • Reynolds number applications
🪂 Sports & Safety
- • Skydiving speed calculations
- • Parachute design
- • BASE jumping analysis
- • Wingsuit flight planning
- • Fall safety assessment
⚙️ Engineering
- • Aerodynamic design
- • Drop testing
- • Atmospheric entry vehicles
- • Seed dispersal studies
- • Particle settling in fluids
📋 Drag Coefficient Reference Table
| Shape/Object | Cd Value | Notes |
|---|---|---|
| Sphere (smooth) | 0.47 | Laminar flow |
| Sphere (rough/golf ball) | 0.25 | Dimples create turbulence |
| Cube | 1.05 | Face perpendicular to flow |
| Cylinder (axis perpendicular) | 1.0 - 1.2 | Depends on length/diameter |
| Flat plate (perpendicular) | 1.28 | Maximum drag |
| Streamlined body | 0.04 - 0.1 | Teardrop shape |
| Human (standing) | 1.0 - 1.3 | Depends on clothing |
| Skydiver (belly-to-earth) | 1.0 | Spread position |
| Skydiver (head-down) | 0.7 | Diving position |
| Parachute (hemispherical) | 1.42 | High drag designed |
| Bicycle + rider | 0.9 - 1.0 | Upright position |
| Car (typical sedan) | 0.25 - 0.35 | Modern aerodynamic |
🌍 Terminal Velocities of Common Objects
| Object | m/s | km/h | mph |
|---|---|---|---|
| 🪶 Feather | 0.5 | 1.8 | 1.1 |
| 🍂 Autumn leaf | 1 | 3.6 | 2.2 |
| ❄️ Snowflake | 1.5 | 5.4 | 3.4 |
| 💧 Raindrop (small) | 4 | 14.4 | 9 |
| 🌧️ Raindrop (large) | 9 | 32.4 | 20 |
| 🏓 Ping pong ball | 10 | 36 | 22 |
| 🐱 Cat | 27 | 97 | 60 |
| ⚾ Baseball | 42 | 151 | 94 |
| 🪂 Skydiver (spread) | 55 | 198 | 123 |
| 🏊 Skydiver (dive) | 90 | 324 | 201 |
| 🦅 Peregrine falcon | 108 | 389 | 242 |
| 🚀 Felix Baumgartner | 373 | 1342 | 834 |
Note: Felix Baumgartner's record was achieved at high altitude with much lower air density
🏔️ Air Density at Different Altitudes
Altitude Effects
- Sea level: ρ = 1.225 kg/m³
- 1,000 m: ρ = 1.112 kg/m³ (91%)
- 3,000 m: ρ = 0.909 kg/m³ (74%)
- 5,000 m: ρ = 0.736 kg/m³ (60%)
- 10,000 m: ρ = 0.414 kg/m³ (34%)
- 12,000 m (Baumgartner): ρ ≈ 0.3 kg/m³ (25%)
- 40,000 m (edge of space): ρ ≈ 0.004 kg/m³ (0.3%)
Why It Matters
- • Lower density = less drag = higher terminal velocity
- • Skydivers at high altitude fall faster initially
- • Parachutes less effective at high altitudes
- • Temperature also affects density
- • Humidity slightly reduces air density
- • Weather conditions change local density
Barometric Formula
Where L = 0.0065 K/m (lapse rate), T₀ = 288.15 K, M = 0.0289644 kg/mol, R = 8.31447 J/(mol·K)
🏆 Famous Record-Breaking Falls
Felix Baumgartner - Red Bull Stratos (2012)
Austrian skydiver who jumped from 39,045 meters (128,100 ft) altitude, reaching a maximum speed of 1,357.64 km/h (843.6 mph) - breaking the sound barrier!
- • Jump altitude: 39,045 m (128,100 ft)
- • Free fall duration: 4 minutes 20 seconds
- • Maximum Mach number: 1.25
- • Air density at jump: ~0.003 kg/m³
Alan Eustace - Stratospheric Jump (2014)
Google executive who broke Baumgartner's altitude record, jumping from 41,425 meters (135,890 ft) using a balloon-launched spacesuit.
- • Jump altitude: 41,425 m (current record)
- • Maximum speed: 1,322 km/h
- • Total freefall time: 4.5 minutes
Peregrine Falcon - Nature's Speed Record
The fastest animal on Earth during its hunting dive (stoop), reaching speeds of over 390 km/h (242 mph).
- • Achieves speed by tucking wings close to body
- • Drag coefficient reduced to approximately 0.1
- • Can pull out of dive at 25+ g-forces
- • Special baffles in nostrils allow breathing at speed
🔬 Practical Applications
🪂 Parachute Design
Parachutes are designed to maximize drag coefficient and area, reducing terminal velocity to a safe landing speed of about 5-7 m/s.
- • Main parachute: landing at ~5 m/s
- • Reserve parachute: landing at ~6-7 m/s
- • Drag chute on aircraft: slowing during landing
- • Drogue chute: stabilizing free fall
🌧️ Meteorology
Terminal velocity of raindrops determines their impact and measurement in rain gauges.
- • Drizzle (0.5mm): ~2 m/s
- • Light rain (1mm): ~4 m/s
- • Moderate rain (2mm): ~6.5 m/s
- • Heavy rain (5mm): ~9 m/s
- • Large drops break up above ~5mm
🚀 Space Reentry
Spacecraft entering atmosphere experience changing terminal velocity as air density increases.
- • Initial speeds: 7-11 km/s
- • Heat shields designed for high drag
- • Drogue chutes deployed at supersonic speeds
- • Main chutes at subsonic speeds
🏭 Industrial Processes
Particle settling in fluids is governed by terminal velocity (Stokes' Law for small particles).
- • Sedimentation tanks in water treatment
- • Particle size analysis
- • Spray drying processes
- • Dust collection systems
❓ Frequently Asked Questions
Q: Can you exceed terminal velocity?
Yes! If you start at high altitude where air is thin and fall into denser air, you can temporarily exceed the terminal velocity for that altitude. Felix Baumgartner exceeded sea-level terminal velocity during his stratospheric jump.
Q: Why do cats survive high falls?
Cats have a low terminal velocity (~27 m/s) due to their ability to spread their legs and arch their backs, increasing drag. They also have a "righting reflex" to land on their feet and flexible bones to absorb impact. Studies show they may actually do better from higher falls as they have time to relax.
Q: What happens to terminal velocity in water?
Terminal velocity is much lower in water due to water's higher density (~1000 kg/m³ vs. 1.225 kg/m³ for air). A human sinks at about 2 m/s in water compared to 55 m/s in air.
Q: Why don't raindrops hurt when they fall?
Raindrops reach terminal velocity within a few meters of falling. Without air resistance, a raindrop from 1 km would hit at ~140 m/s (500 km/h)! Terminal velocity limits them to 4-9 m/s depending on size.
Q: How does a skydiver control their speed?
By changing body position: spreading out increases drag and slows down, while tucking into a ball or diving head-down reduces drag and speeds up. Skilled skydivers can vary their speed from ~55 m/s to ~90 m/s just by body position.
Q: Does weight affect terminal velocity?
Yes, heavier objects have higher terminal velocities when other factors are equal. The formula shows v_t is proportional to √m. A 100 kg skydiver falls about 12% faster than an 80 kg skydiver in the same position.
Q: What is the terminal velocity of a human on Mars?
Mars has about 1% of Earth's atmospheric density. A human would have a terminal velocity of about 500+ m/s on Mars - much too fast for a safe landing! This is why Mars landers need heat shields, parachutes, AND rockets to land safely.
Q: How long does it take to reach terminal velocity?
Most objects reach 99% of terminal velocity in about 5 time constants (τ = m/ρCdAv_t). For a skydiver, this is about 12-15 seconds. Lighter objects with high drag reach terminal faster - a feather reaches terminal almost instantly.
Q: What is the fastest terminal velocity ever achieved by a human?
Alan Eustace reached 1,322 km/h (822 mph) during his 2014 stratospheric jump from 41,425 meters. At that altitude, air density is less than 0.3% of sea level, allowing for extremely high speeds during the initial phase of the fall.
💡 Key Takeaways
Physics Principles
- ✓ Terminal velocity occurs when drag equals weight
- ✓ Drag force is proportional to velocity squared
- ✓ Shape matters more than mass for air resistance
- ✓ Lower density = higher terminal velocity
- ✓ Velocity approaches terminal asymptotically
Practical Facts
- ✓ Skydivers reach terminal in 10-15 seconds
- ✓ Body position can change speed by 40%
- ✓ Parachutes reduce terminal to safe landing speed
- ✓ Cats survive high falls due to low terminal velocity
- ✓ Raindrops are limited to 9 m/s by terminal velocity
❓ Frequently Asked Questions
Q: Why can't objects accelerate beyond terminal velocity?
At terminal velocity, air drag exactly equals weight, so net force is zero. With zero net force, there's no acceleration (Newton's Second Law). The object continues at constant velocity.
Q: How do parachutes work?
Parachutes dramatically increase cross-sectional area and drag coefficient. This lowers terminal velocity from ~55 m/s (skydiver) to about 5-6 m/s (with parachute), allowing safe landing.
Q: Does a heavier object fall faster?
For terminal velocity, yes! Heavier objects have higher terminal velocity because they need more drag to balance their greater weight. Without air resistance, all objects fall at the same rate.
📝 Key Takeaways
- • Terminal velocity: v_t = √(2mg / (ρAC_d))
- • Occurs when drag force equals weight
- • Drag force ∝ velocity² (at high speeds)
- • Lower air density = higher terminal velocity
- • Streamlined shapes have lower drag coefficients
- • Human skydiver: ~55 m/s (flat) to ~90 m/s (head down)
📊 Terminal Velocities Comparison
| Object | Terminal Velocity | Notes |
|---|---|---|
| Raindrop | 9 m/s | Size dependent |
| Cat | 27 m/s | Can survive |
| Skydiver (flat) | 55 m/s | Belly to earth |
| Skydiver (dive) | 90 m/s | Head down |
🔢 Quick Formulas
v_t = √(2mg / ρAC_d)
F_drag = ½ρv²AC_d
At terminal: F_drag = mg
Time to 95%: t ≈ v_t/g × 3
💡 Interesting Fact
A skydiver reaches 95% of terminal velocity in about 12-15 seconds. The last 5% takes much longer due to exponential approach!
❓ Frequently Asked Questions
Q: Can you exceed terminal velocity?
Yes! If you start at high altitude where air is thin and fall into denser air, you can temporarily exceed the terminal velocity for that altitude. Felix Baumgartner exceeded sea-level terminal velocity during his stratospheric jump.
Q: Why do cats survive high falls?
Cats have a low terminal velocity (~27 m/s) due to their ability to spread their legs and arch their backs, increasing drag. They also have a "righting reflex" to land on their feet and flexible bones to absorb impact.
Q: What happens to terminal velocity in water?
Terminal velocity is much lower in water due to water's higher density (~1000 kg/m³ vs. 1.225 kg/m³ for air). A human sinks at about 2 m/s in water compared to 55 m/s in air.
Q: Why don't raindrops hurt when they fall?
Raindrops reach terminal velocity within a few meters of falling. Without air resistance, a raindrop from 1 km would hit at ~140 m/s (500 km/h)! Terminal velocity limits them to 4-9 m/s depending on size.
Q: How does a skydiver control their speed?
By changing body position: spreading out increases drag and slows down, while tucking into a ball or diving head-down reduces drag and speeds up. Skilled skydivers can vary their speed from ~55 m/s to ~90 m/s just by body position.
Q: Does weight affect terminal velocity?
Yes, heavier objects have higher terminal velocities when other factors are equal. The formula shows v_t is proportional to √m. A 100 kg skydiver falls about 12% faster than an 80 kg skydiver in the same position.
Q: What is the terminal velocity of a human on Mars?
Mars has about 1% of Earth's atmospheric density. A human would have a terminal velocity of about 500+ m/s on Mars - much too fast for a safe landing! This is why Mars landers need heat shields, parachutes, AND rockets to land safely.
Q: How long does it take to reach terminal velocity?
Most objects reach 99% of terminal velocity in about 5 time constants (τ = m/ρCdAv_t). For a skydiver, this is about 12-15 seconds. Lighter objects with high drag reach terminal faster - a feather reaches terminal almost instantly.
Official Data Sources
Disclaimer
This calculator provides theoretical calculations based on simplified drag models. Actual terminal velocities may vary due to turbulence, non-uniform air density, object rotation, and other factors not included in the basic drag equation. The calculations assume quadratic drag (F_d = ½ρv²CdA) which is valid for most objects at typical speeds, but may not apply at very low speeds (Stokes' law) or supersonic speeds. Drag coefficients are approximate and depend on Reynolds number, surface roughness, and object orientation. Always verify critical values for safety-critical applications like skydiving or engineering design. Results are for educational purposes only.
For educational and informational purposes only. Verify with a qualified professional.
🔬 Physics Facts
Skydiver terminal velocity ~55 m/s (120 mph)
— NASA
Drag force F_d = ½ρv²AC_d
— Fluid Dynamics
Raindrop ~9 m/s due to small size
— Meteorology
Parachute reduces v_term to ~5 m/s
— Aerodynamics
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