Terminal Velocity Calculator

General Physics

This tool calculates the maximum constant speed a freely falling object eventually reaches when the resistance of the medium (such as air) through which it is falling prevents further acceleration.

Practical Examples

Explore different scenarios to understand how terminal velocity works in the real world.

Average Skydiver

Skydiver

Calculates the terminal velocity of an average-sized skydiver in a belly-to-earth position.

Mass: 75 kg, Area: 0.7 m²

Drag Coeff: 1.0, Fluid Density: 1.225 kg/m³

Large Raindrop

Raindrop

Determines the maximum speed of a large raindrop falling through the air.

Mass: 0.0000335 kg, Area: 0.0000196 m²

Drag Coeff: 0.6, Fluid Density: 1.225 kg/m³

Falling Bowling Ball

Bowling Ball

Finds the terminal velocity of a standard bowling ball dropped from a great height.

Mass: 7.26 kg, Area: 0.0366 m²

Drag Coeff: 0.45, Fluid Density: 1.225 kg/m³

Large Hailstone

Hailstone

Computes the terminal velocity for a large, golf-ball-sized hailstone.

Mass: 0.02 kg, Area: 0.001257 m²

Drag Coeff: 0.5, Fluid Density: 1.225 kg/m³

Other Titles
Understanding Terminal Velocity: A Comprehensive Guide
Delve into the core concepts of terminal velocity, from the underlying physics to its real-world significance and calculations.

What is Terminal Velocity?

  • The Balance of Forces
  • Factors Influencing Terminal Velocity
  • Reaching a Constant Speed
Terminal velocity is the highest velocity attainable by an object as it falls through a fluid (air is the most common example). It occurs at the point when the downward force of gravity is exactly balanced by the upward force of drag (air resistance). At this stage, the object's net force is zero, meaning it stops accelerating and continues to fall at a constant speed.
The Role of Gravity and Drag
When an object first starts to fall, its velocity is low, and the force of air resistance is small. Because the force of gravity is much stronger, the object accelerates downwards. As its speed increases, so does the air resistance. This continues until the force of drag becomes equal in magnitude to the force of gravity. This equilibrium point defines the terminal velocity.

Step-by-Step Guide to Using the Terminal Velocity Calculator

  • Inputting Your Data
  • Interpreting the Results
  • Using the Examples
Our calculator simplifies the process of finding terminal velocity. Follow these steps for an accurate calculation:
Field-by-Field Instructions
  1. Mass (m): Enter the object's mass in kilograms (kg). A heavier object will have a stronger gravitational pull, generally leading to a higher terminal velocity.
  2. Cross-Sectional Area (A): Input the object's projected area in square meters (m²). This is the area of the object's silhouette from the perspective of the fluid it's moving through. A larger area increases drag.
  3. Drag Coefficient (Cd): This dimensionless number represents how aerodynamic the object is. A lower value means it's more streamlined. For instance, a sphere has a Cd of about 0.47, while a skydiver in a belly-flop position has a Cd closer to 1.0.
  4. Fluid Density (ρ): Enter the density of the fluid in kilograms per cubic meter (kg/m³). The default value, 1.225 kg/m³, is the approximate density of air at sea level. The denser the fluid, the greater the resistance.

Real-World Applications of Terminal Velocity

  • Skydiving and Parachuting
  • Vehicle and Aircraft Design
  • Natural Phenomena
The concept of terminal velocity is not just an academic exercise; it has crucial applications in various fields.
Engineering and Safety
In skydiving, controlling terminal velocity is key. A skydiver changes their body position to manipulate their cross-sectional area and drag coefficient, thereby controlling their descent speed. A parachute dramatically increases the area and drag, slowing the descent to a safe landing speed. In automotive and aerospace engineering, designers aim to minimize the drag coefficient to improve fuel efficiency and performance.
Nature's Design
Terminal velocity governs why raindrops don't hit us at lethal speeds. A small raindrop reaches its terminal velocity of about 9 m/s (20 mph) quickly. Animals, like cats, can often survive falls from great heights because their low mass and relatively large surface area result in a lower terminal velocity.

Common Misconceptions and Correct Methods

  • Heavier Objects Fall Faster?
  • The Myth of the Vacuum
  • Is Terminal Velocity Reached Instantly?
Aristotle famously (and incorrectly) thought that heavier objects fall proportionally faster than lighter ones. In a vacuum, all objects fall at the same rate of acceleration regardless of their mass (as demonstrated by Galileo). However, in the presence of air resistance, the situation changes. While mass is a factor in the terminal velocity equation, so are area and shape. A heavy but very non-aerodynamic object might have a lower terminal velocity than a lighter, more streamlined object.
Acceleration is Not Constant
A falling object does not instantly reach terminal velocity. It begins by accelerating at approximately the rate of gravity (g ≈ 9.81 m/s²). As speed builds, the opposing drag force increases, and the rate of acceleration decreases. The object only reaches terminal velocity when this acceleration becomes zero.

Mathematical Derivation and Examples

  • The Core Formula
  • Deriving the Equation
  • Manual Calculation Walkthrough
The calculation is based on the equilibrium of forces.
The Formula: Vt = √((2 m g) / (ρ A Cd))
  • Vt is the terminal velocity.
  • m is the mass of the object.
  • g is the acceleration due to gravity (≈ 9.81 m/s²).
  • ρ (rho) is the density of the fluid.
  • A is the projected cross-sectional area.
  • Cd is the drag coefficient.
Example Calculation
Let's calculate the terminal velocity for the skydiver example: m = 75 kg, A = 0.7 m², Cd = 1.0, ρ = 1.225 kg/m³. We use g = 9.81 m/s².

Calculation Steps:

  • Force of Gravity (Fg) = m * g = 75 * 9.81 = 735.75 N
  • At terminal velocity, Drag Force (Fd) = Fg.
  • Fd = 0.5 * ρ * A * Cd * Vt²
  • 735.75 = 0.5 * 1.225 * 0.7 * 1.0 * Vt²
  • 735.75 = 0.42875 * Vt²
  • Vt² = 735.75 / 0.42875 = 1716.0
  • Vt = √1716.0 ≈ 41.42 m/s