🪂 Free Fall with Air Resistance Calculator
Calculate falling with air resistance
How to Use This Calculator
Enter Height
Input the height (h) in meters from which the object falls. This is the vertical distance above the ground.
Enter Mass
Enter the object's mass (m) in kilograms. Heavier objects have higher terminal velocities, as weight increases faster than drag with mass.
Enter Drag Coefficient
Input the drag coefficient (C_d), typically 0.47 for spheres, 0.5-1.0 for irregular shapes. Streamlined objects have lower C_d values (0.04-0.1).
Enter Cross-Sectional Area
Enter the frontal area (A) in m² that faces the direction of motion. Larger areas experience more drag. For spheres, A = πr².
Enter Air Density
Input air density (ρ) in kg/m³. Standard sea level = 1.225 kg/m³. Higher altitude = lower density (less drag). Default is 1.225.
Click Calculate
Press "Calculate" to find terminal velocity, fall time, and final velocity accounting for air resistance.
Formula
v_t = √(2mg / (C_d × ρ × A))
F_drag = ½ × C_d × ρ × A × v²
k = (C_d × ρ × A) / (2m)
Where:
- v_t = Terminal velocity (m/s) - maximum falling speed
- m = Mass (kg)
- g = Acceleration due to gravity (9.81 m/s²)
- C_d = Drag coefficient (dimensionless)
- ρ = Air density (kg/m³)
- A = Cross-sectional area (m²)
- F_drag = Drag force (N)
- k = Air resistance constant
Example Calculation:
A skydiver (m=70 kg, C_d=0.5, A=0.7 m²) falling from 3000 m:
1. Terminal velocity: v_t = √(2 × 70 × 9.81 / (0.5 × 1.225 × 0.7)) = √(1373 / 0.429) = 56.5 m/s (203 km/h)
2. At terminal velocity, drag force equals weight: F_drag = mg = 687 N
Note: Actual fall time and velocity depend on how quickly terminal velocity is reached, which requires solving differential equations.
About Free Fall with Air Resistance Calculator
The Free Fall with Air Resistance Calculator determines how objects fall when air drag is significant. Unlike simple free fall, air resistance limits maximum falling speed to terminal velocity, where drag force balances weight. This calculator is essential for understanding real-world falling motion, skydiving, parachuting, and any scenario where air resistance matters.
Terminal Velocity
Terminal velocity is the maximum speed an object reaches when falling. At terminal velocity, drag force equals weight (mg), so net force is zero and acceleration stops. Terminal velocity depends on mass, drag coefficient, cross-sectional area, and air density. Heavier, denser, or more streamlined objects have higher terminal velocities.
Examples: Skydiver ≈ 50-60 m/s (180-220 km/h), Baseball ≈ 42 m/s (150 km/h), Raindrop ≈ 9 m/s (32 km/h), Parachutist ≈ 5-7 m/s (18-25 km/h).
Air Resistance (Drag)
Air resistance (drag) is proportional to the square of velocity: F_drag = ½C_dρAv². This means doubling speed quadruples drag. As an object falls and speeds up, drag increases until it balances weight, reaching terminal velocity. Before terminal velocity, the object accelerates; after reaching it, speed remains constant.
Key Factors
- Mass: Heavier objects have higher terminal velocities because weight (mg) increases with mass, while drag increases with area. Mass appears in the numerator of terminal velocity formula.
- Cross-Sectional Area: Larger frontal area increases drag, reducing terminal velocity. Skydivers spread out to increase area and slow down; they streamline to decrease area and fall faster.
- Drag Coefficient: Shape matters! Streamlined objects (low C_d ≈ 0.04) fall much faster than blunt objects (C_d ≈ 0.5-1.0) of the same size and weight.
- Air Density: Thinner air (high altitude) reduces drag, increasing terminal velocity. At 10 km altitude, terminal velocity is about 10% higher than at sea level.
Practical Applications
- Skydiving: Calculate terminal velocities and fall times for different body positions and equipment configurations.
- Parachute Design: Design parachutes with appropriate sizes and shapes to achieve safe landing speeds.
- Sports: Understand how air resistance affects falling sports equipment, determining optimal shapes and sizes.
- Safety Engineering: Design safety systems and calculate impact speeds for objects falling from heights, accounting for air resistance effects.
- Meteorology: Model raindrop and hailstone falling speeds, important for weather prediction and understanding precipitation.
Understanding the Results
Terminal Velocity: The maximum falling speed when drag balances weight. Objects approach this speed asymptotically, reaching ~99% of terminal velocity after falling about 10-15 times the distance needed to reach it in vacuum.
Fall Time: Longer than free fall time due to air resistance slowing acceleration. The difference increases with height and decreases with object density.
Final Velocity: For long falls, final velocity approaches terminal velocity. For short falls, it may be significantly less than terminal velocity.
Frequently Asked Questions
What is terminal velocity?
Terminal velocity is the constant maximum speed reached by a falling object when drag force equals weight. At this speed, net force is zero, so acceleration stops. The object continues falling at constant speed rather than accelerating indefinitely.
How does air resistance change with speed?
Drag force is proportional to velocity squared (F ∝ v²). This means drag increases rapidly with speed - doubling speed quadruples drag. That's why objects reach terminal velocity instead of accelerating forever - drag grows until it matches weight.
Why do heavier objects fall faster with air resistance?
Weight increases with mass (mg), but drag depends mainly on area. Heavier objects have more weight per unit area, so they need higher speeds for drag to balance weight. A bowling ball falls faster than a tennis ball because it's much denser (same size, more mass).
How do skydivers control their fall speed?
By changing their cross-sectional area! Spreading out (belly-to-earth) increases area and drag, reducing terminal velocity to ~50 m/s. Diving headfirst (streamlined) decreases area, increasing terminal velocity to ~90+ m/s. Parachutes dramatically increase area, reducing speed to ~5-7 m/s.
Does air density really matter?
Yes! Air density at 10 km altitude is about 40% of sea level density. This means less drag, so terminal velocity is higher. Skydivers fall faster at higher altitudes. For precise calculations, especially at altitude, use appropriate air density values.
How long does it take to reach terminal velocity?
Objects reach about 95% of terminal velocity after falling roughly 3-5 times the distance they would fall in vacuum to reach that speed. For a typical skydiver, this is about 10-15 seconds or 300-500 meters. Exact time depends on the object's characteristics.
When can I ignore air resistance?
Air resistance is negligible for dense, compact objects falling short distances. For example, a steel ball falling 10 meters experiences minimal air resistance. For light objects (feathers), large areas, or long falls (skydiving), air resistance is essential and must be included.