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Terminal Velocity Formula Calculator

\[ v_t = \sqrt{\frac{2mg}{\rho A C_d}} \]

Unitless

1. What is the Terminal Velocity Formula Calculator?

Definition: This calculator computes the terminal velocity (\(v_t\)) of an object falling through a fluid using the formula \( v_t = \sqrt{\frac{2mg}{\rho A C_d}} \), where \( m \) is the mass, \( g \) is the gravitational acceleration, \( \rho \) is the fluid density, \( A \) is the cross-sectional area, and \( C_d \) is the drag coefficient.

Purpose: It is used in physics and engineering to determine the maximum velocity of a falling object when gravitational and drag forces balance, applicable in scenarios like skydiving, raindrop motion, and particle settling.

2. How Does the Calculator Work?

The calculator uses the terminal velocity formula:

Formula: \[ v_t = \sqrt{\frac{2mg}{\rho A C_d}} \] where:

  • \(v_t\): Terminal velocity (m/s, ft/s)
  • \(m\): Mass (kg, lb)
  • \(g\): Gravitational acceleration (m/s², ft/s²)
  • \(\rho\): Air density (kg/m³, lb/ft³)
  • \(A\): Cross-sectional area (m², ft²)
  • \(C_d\): Drag coefficient (unitless)

Unit Conversions:

  • Mass (\(m\)):
    • 1 kg = 1 kg
    • 1 lb = 0.45359237 kg
  • Gravitational Acceleration (\(g\)):
    • 1 m/s² = 1 m/s²
    • 1 ft/s² = 0.3048 m/s²
  • Air Density (\(\rho\)):
    • 1 kg/m³ = 1 kg/m³
    • 1 lb/ft³ = 16.01846337396 kg/m³
  • Area (\(A\)):
    • 1 m² = 1 m²
    • 1 ft² = 0.09290304 m²
  • Terminal Velocity (Output):
    • 1 m/s = 1 m/s
    • 1 ft/s = 0.3048 m/s
The terminal velocity is calculated in m/s and can be converted to the selected output unit (m/s, ft/s). Results greater than 10,000 or less than 0.001 are displayed in scientific notation; otherwise, they are shown with 4 decimal places.

Steps:

  • Enter the mass (\(m\)), gravitational acceleration (\(g\)), air density (\(\rho\)), cross-sectional area (\(A\)), and drag coefficient (\(C_d\)) with their units (default: \(m = 80 \, \text{kg}\), \(g = 9.81 \, \text{m/s}^2\), \(\rho = 1.225 \, \text{kg/m}^3\), \(A = 0.7 \, \text{m}^2\), \(C_d = 1\)).
  • Convert inputs to SI units (kg, m/s², kg/m³, m²).
  • Validate that all inputs are greater than 0.
  • Calculate the terminal velocity in m/s using the formula.
  • Convert the terminal velocity to the selected output unit.
  • Display the result, using scientific notation if the value is greater than 10,000 or less than 0.001, otherwise rounded to 4 decimal places.

3. Importance of Terminal Velocity Calculation

Calculating terminal velocity is crucial for:

  • Physics: Understanding the motion of objects in fluids, such as raindrops, skydivers, or particles in sedimentation processes, where drag balances gravity.
  • Engineering: Designing parachutes, aircraft, and vehicles, where terminal velocity affects safety and performance (e.g., a skydiver’s descent speed).
  • Education: Teaching the principles of fluid dynamics, drag forces, and the balance of forces in free fall.

4. Using the Calculator

Examples:

  • Example 1: Calculate the terminal velocity for a skydiver with \( m = 80 \, \text{kg} \), \( g = 9.81 \, \text{m/s}^2 \), \( \rho = 1.225 \, \text{kg/m}^3 \), \( A = 0.7 \, \text{m}^2 \), \( C_d = 1 \), output in m/s:
    • Enter \( m = 80 \, \text{kg} \), \( g = 9.81 \, \text{m/s}^2 \), \( \rho = 1.225 \, \text{kg/m}^3 \), \( A = 0.7 \, \text{m}^2 \), \( C_d = 1 \).
    • Numerator: \( 2mg = 2 \times 80 \times 9.81 = 1569.6 \, \text{N} \).
    • Denominator: \( \rho A C_d = 1.225 \times 0.7 \times 1 = 0.8575 \, \text{kg/m} \).
    • Fraction: \( \frac{2mg}{\rho A C_d} = \frac{1569.6}{0.8575} \approx 1830.0292 \, \text{m}^2/\text{s}^2 \).
    • Terminal velocity: \( v_t = \sqrt{1830.0292} \approx 42.7791 \, \text{m/s} \).
    • Output unit: m/s (no conversion needed).
    • Result: \( \text{Terminal Velocity} = 42.7791 \, \text{m/s} \).
  • Example 2: Calculate the terminal velocity for an object with \( m = 176.37 \, \text{lb} \), \( g = 32.174 \, \text{ft/s}^2 \), \( \rho = 0.0765 \, \text{lb/ft}^3 \), \( A = 7.5347 \, \text{ft}^2 \), \( C_d = 1 \), output in ft/s:
    • Enter \( m = 176.37 \, \text{lb} \), \( g = 32.174 \, \text{ft/s}^2 \), \( \rho = 0.0765 \, \text{lb/ft}^3 \), \( A = 7.5347 \, \text{ft}^2 \), \( C_d = 1 \).
    • Convert: \( m = 176.37 \times 0.45359237 = 80 \, \text{kg} \), \( g = 32.174 \times 0.3048 = 9.81 \, \text{m/s}^2 \), \( \rho = 0.0765 \times 16.01846337396 = 1.225 \, \text{kg/m}^3 \), \( A = 7.5347 \times 0.09290304 = 0.7 \, \text{m}^2 \).
    • Numerator: \( 2mg = 2 \times 80 \times 9.81 = 1569.6 \, \text{N} \).
    • Denominator: \( \rho A C_d = 1.225 \times 0.7 \times 1 = 0.8575 \, \text{kg/m} \).
    • Fraction: \( \frac{2mg}{\rho A C_d} = \frac{1569.6}{0.8575} \approx 1830.0292 \, \text{m}^2/\text{s}^2 \).
    • Terminal velocity in m/s: \( v_t = \sqrt{1830.0292} \approx 42.7791 \, \text{m/s} \).
    • Convert to output unit (ft/s): \( 42.7791 \times \frac{1}{0.3048} \approx 140.3514 \, \text{ft/s} \).
    • Result: \( \text{Terminal Velocity} = 140.3514 \, \text{ft/s} \).

5. Frequently Asked Questions (FAQ)

Q: What is terminal velocity?
A: Terminal velocity (\(v_t\)) is the constant maximum velocity reached by an object falling through a fluid (e.g., air) when the gravitational force (\( mg \)) is balanced by the drag force (\( \frac{1}{2} \rho v_t^2 A C_d \)), resulting in zero acceleration. It is given by \( v_t = \sqrt{\frac{2mg}{\rho A C_d}} \).

Q: Why must all inputs be greater than zero?
A: All inputs must be greater than zero to represent physical quantities: mass, gravitational acceleration, air density, area, and drag coefficient must be positive for the object to exist and experience drag. A zero value in the denominator (\( \rho A C_d \)) would lead to division by zero, making the calculation undefined.

Q: How does the drag coefficient affect terminal velocity?
A: The drag coefficient (\(C_d\)) depends on the object’s shape and surface. A higher \(C_d\) (e.g., a parachute with \(C_d \approx 1.5\)) increases drag, reducing terminal velocity. A lower \(C_d\) (e.g., a streamlined object with \(C_d \approx 0.04\)) decreases drag, increasing terminal velocity. For example, a skydiver with a parachute falls slower than one without due to a higher \(C_d\).

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