Free Fall with Air Resistance

General Physics

This calculator determines the key metrics of an object in free fall, including the effects of air resistance (drag).

Practical Examples

Use these pre-filled examples to see how the calculator works in different scenarios.

Average Skydiver

Skydiver

An average-sized skydiver (75 kg) falling in a stable, belly-to-earth position.

Mass: 75 kg, Drag Coeff: 0.7

Area: 0.7 m², Time: 10 s, Air Density: 1.225 kg/m³

Falling Basketball

Basketball

A standard basketball dropped from a height.

Mass: 0.625 kg, Drag Coeff: 0.47

Area: 0.045 m², Time: 3 s, Air Density: 1.225 kg/m³

Large Raindrop

Raindrop

A large, spherical raindrop falling from the clouds.

Mass: 0.0000335 kg, Drag Coeff: 0.47

Area: 0.0000126 m², Time: 60 s, Air Density: 1.225 kg/m³

Bowling Ball

Bowling Ball

A heavy bowling ball, where air resistance has a smaller effect relative to its weight.

Mass: 7 kg, Drag Coeff: 0.47

Area: 0.0366 m², Time: 5 s, Air Density: 1.225 kg/m³

Other Titles
Understanding the Free Fall with Air Resistance Calculator
An in-depth guide to the physics of falling objects when air resistance is a factor, exploring concepts like terminal velocity and drag force.

The Physics of Falling: Beyond the Vacuum

  • What is Free Fall?
  • The Role of Air Resistance (Drag)
  • Reaching Terminal Velocity
In a perfect vacuum, all objects fall at the same rate due to gravity. However, on Earth, air resistance (or drag) opposes this motion. The Free Fall with Air Resistance Calculator models this real-world scenario, providing a more accurate picture of how objects fall through our atmosphere.
Key Forces at Play
Two primary forces act on a falling object: Gravity (pulling it down) and Air Resistance (pushing it up). Gravity is constant (F = mg), but air resistance increases with velocity. The calculator uses the drag equation: F_d = 0.5 ρ v² Cd A.
When the upward force of air resistance equals the downward force of gravity, the net force is zero. The object stops accelerating and falls at a constant speed known as Terminal Velocity.

Step-by-Step Guide to Using the Calculator

  • Inputting Your Data Correctly
  • Understanding the Output Metrics
  • Leveraging the Examples
Input Fields Explained
1. Mass (m): The object's mass in kilograms. Heavier objects have a stronger gravitational pull.
2. Drag Coefficient (Cd): A dimensionless number representing the object's aerodynamic efficiency. A lower number means it's more streamlined.
3. Cross-Sectional Area (A): The object's area facing the direction of fall, in square meters. A larger area catches more air, increasing drag.
4. Time (t): The duration of the fall in seconds for which you want to calculate the metrics.
5. Air Density (ρ): The density of the air in kg/m³. This value changes with altitude and temperature. 1.225 kg/m³ is the standard at sea level.
Interpreting the Results
  • Terminal Velocity: The maximum speed the object can reach in free fall.\n- Velocity at Time (t): The object's speed at the specific time you entered.\n- Distance Fallen: How far the object has fallen during that time.\n- Drag Force at Time (t): The magnitude of the air resistance force at that specific time and velocity.

Real-World Applications and Examples

  • Skydiving and Parachuting
  • Automotive and Aerospace Engineering
  • Meteorology and Ballistics
Understanding free fall with drag is crucial in many fields.
Applications
Skydiving: A skydiver changes their cross-sectional area to control their fall speed, then deploys a parachute to drastically increase drag and land safely.\nEngineering: Car and airplane designers aim to minimize the drag coefficient to improve fuel efficiency and speed.\nMeteorology: The size and terminal velocity of raindrops and hailstones are determined by these physical principles.

Mathematical Derivation and Formulas

  • The Equation of Motion
  • Deriving Terminal Velocity
  • Calculating Velocity and Position Over Time
The Core Equation
The net force on the object is Fnet = Fgravity - F_drag. Using Newton's second law (F=ma), we get the differential equation: m dv/dt = mg - 0.5 ρ v² Cd * A.
Formula for Terminal Velocity (v_t)
At terminal velocity, acceleration is zero, so mg = 0.5 ρ vt² Cd A. Solving for vt gives: v_t = sqrt((2 mg) / (ρ Cd * A)).
Velocity and Distance Formulas
Solving the differential equation yields the velocity as a function of time: v(t) = vt tanh((g t) / vt). Integrating this equation gives the distance fallen: d(t) = (vt² / g) ln(cosh((g t) / vt)).

Common Misconceptions and Key Factors

  • Heavier Objects Don't Always Fall Faster
  • The Myth of Instant Terminal Velocity
  • Altitude's Impact on Air Density
Mass vs. Shape
While a heavier object experiences a greater gravitational force, its terminal velocity is a balance between its mass and its shape (drag). A light but very aerodynamic object could have a higher terminal velocity than a heavy but un-aerodynamic one. Compare a steel ball to a flat sheet of steel of the same mass.
Acceleration is Not Constant
An object does not instantly reach terminal velocity. It starts accelerating at 'g' (9.8 m/s²) and the acceleration decreases as velocity and drag increase, eventually reaching zero at terminal velocity. This calculator shows the velocity at any point along this curve.
Furthermore, as an object falls from a great height, air density increases, which means the drag force and terminal velocity are not constant but change during the fall. This calculator assumes a constant air density for simplicity.