The **terminal velocity equation** provides a means to calculate the maximum velocity reached by a falling object when the upward force of air resistance equals the downward force of gravity. It can be expressed as v_{t} = √2 m g/ρ A C_{D}, where v_{t} represents the terminal velocity, m is the mass of the object, g is the gravitational acceleration, ρ is the density of the medium through which the object is falling, A is the cross-sectional area of the object, and C_{D} is the drag coefficient.

## Practice problems

### Problem #1

What is the terminal velocity of a 0.5 kg parachute bag thrown from a plane? Given the density of the air as 1.21 kg/m^{3}, the cross-sectional area of the parachute bag as 0.10 m^{2}, and the drag coefficient as 0.12. Take the value of gravitational acceleration as g = 9.81 m/s^{2}.

**Solution**

Given data:

- Terminal velocity of a parachute bag, v
_{t}= ? - Mass of a parachute bag, m = 0.5 kg
- Density of the air, ρ = 1.21 kg/m
^{3} - Cross sectional area of a parachute bag, A = 0.10 m
^{2} - Drag coefficient, C
_{D}= 0.12 - Gravitational acceleration, g = 9.81 m/s
^{2}

Applying the formula:

- v
_{t}= √2 m g/ρ A C_{D} - v
_{t}= √(2 × 0.5 × 9.81)/(1.21 × 0.10 × 0.12) - v
_{t}= √(9.81)/(0.0145) - v
_{t}= √676.5517 - v
_{t}= 26.01 m/s

Therefore, the terminal velocity of a parachute bag is **26.01 m/s**.

### Problem #2

When a skydiver weighing 55 kg jumps from a helicopter, he reaches the terminal velocity after traveling a certain distance. Calculate the terminal velocity of the skydiver, considering the density of the air as 1.25 kg/m^{3}, the cross-sectional area of the skydiver as 0.20 m^{2}, and the drag coefficient as 0.70.

**Solution**

Given data:

- Mass of a skydiver, m = 55 kg
- Terminal velocity of a skydiver, v
_{t}= ? - Density of the air, ρ = 1.25 kg/m
^{3} - Cross sectional area of a skydiver, A = 0.20 m
^{2} - Drag coefficient, C
_{D}= 0.70 - Assuming gravitational acceleration, g = 9.81 m/s
^{2}

Applying the formula:

- v
_{t}= √2 m g/ρ A C_{D} - v
_{t}= √(2 × 55 × 9.81)/(1.25 × 0.20 × 0.70) - v
_{t}= √(1079.1)/(0.175) - v
_{t}= √6166.2857 - v
_{t}= 78.52 m/s

Therefore, the terminal velocity of a skydiver is **78.52 m/s**.

### Problem #3

A leaf with a mass of 0.05 kg is floating in the air and descending from a certain height, eventually reaching its terminal velocity “vt.” If the density of the air is 1.22 kg/m^{3}, the cross-sectional area of the leaf is 0.05 m^{2}, and the drag coefficient is 0.08, determine the value of the leaf’s terminal velocity.

**Solution**

Given data:

- Mass of a leaf, m = 0.05 kg
- Density of the air, ρ = 1.22 kg/m
^{3} - Cross sectional area of a leaf, A = 0.05 m
^{2} - Drag coefficient, C
_{D}= 0.08 - Terminal velocity of a leaf, v
_{t}= ? - Assuming gravitational acceleration, g = 9.81 m/s
^{2}

Applying the formula:

- v
_{t}= √2 m g/ρ A C_{D} - v
_{t}= √(2 × 0.05 × 9.81)/(1.22 × 0.05 × 0.08) - v
_{t}= √(0.981)/(0.0048) - v
_{t}= √204.375 - v
_{t}= 14.29 m/s

Therefore, the terminal velocity of a leaf is **14.29 m/s**.

### Problem #4

Calculate the terminal velocity of a 0.012 kg feather floating in the air. Given the density of the air as 1.20 kg/m^{3}, the cross-sectional area of the feather as 0.02 m^{2}, and the drag coefficient as 0.05. Take the value of gravitational acceleration as g = 9.81 m/s^{2}.

**Solution**

Given data:

- Terminal velocity of a feather, v
_{t}= ? - Mass of a feather, m = 0.012 kg
- Density of the air, ρ = 1.20 kg/m
^{3} - Cross sectional area of a feather, A = 0.02 m
^{2} - Drag coefficient, C
_{D}= 0.05 - Gravitational acceleration, g = 9.81 m/s
^{2}

Applying the formula:

- v
_{t}= √2 m g/ρ A C_{D} - v
_{t}= √(2 × 0.012 × 9.81)/(1.20 × 0.02 × 0.05) - v
_{t}= √(0.2354)/(0.0012) - v
_{t}= √196.1666 - v
_{t}= 14 m/s

Therefore, the terminal velocity of a feather is **14 m/s**.

## Related

- Free fall equation
- Air resistance formula
**Terminal velocity equation**- Gravity equation
- Inertia formula
- Acceleration formula
- Momentum equation
- Mass formula
- Velocity formula
- Pressure equation
- Kinematic equations
- Friction equation

## External links

- Terminal Velocity – NASA (.gov)
- 3 Ways to Calculate Terminal Velocity – wikiHow
- Terminal Velocity Calculator – Omni Calculator
- Terminal Velocity Definition, Formula & Examples – Study.com
- 6.7: Drag Force and Terminal Speed – Physics LibreTexts
- Drag Force and Terminal Speed – BCcampus Pressbooks
- Terminal velocity – Wikipedia
- Terminal Velocity Calculator – GIGACalculator.com
- Terminal Velocity – HyperPhysics Concepts
- How to Calculate Terminal Velocity – Sciencing
- How can I calculate the terminal velocity of a falling object? – Quora
- The Difference Between Terminal Velocity and Free Fall – ThoughtCo
- 6.4 Drag Force and Terminal Speed – UCF Pressbooks
- Is there a way to calculate the time taken by a falling object to reach terminal velocity? – Physics Stack Exchange
- Terminal Velocity – an overview – ScienceDirect
- Terminal Velocity: Definition, Formula, Speed, Calculate – Vaia
- Drag Force and Terminal Velocity – AK Lectures
- Drag – University of Tennessee, Knoxville
- Terminal Velocity Calculator – CalcTool
- Terminal Velocity Calculator – Calculator Academy
- Terminal Velocity Calculator – GoSkydive
- What is Terminal Velocity? How Do We Find It? – Flipping Physics
- Terminal Velocity of a Human, Free Fall and Drag Force – Owlcation
- Terminal velocity | Definition, Examples, & Facts – Britannica
- Terminal Velocity Calculator – Calculator-online.net
- Which is the equation for terminal velocity? – Brainly

Deep

Learnool.com was founded by Deep Rana, who is a mechanical engineer by profession and a blogger by passion. He has a good conceptual knowledge on different educational topics and he provides the same on this website. He loves to learn something new everyday and believes that the best utilization of free time is developing a new skill.