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

- https://www.wikihow.com/Calculate-Terminal-Velocity
- https://www.omnicalculator.com/physics/terminal-velocity
- https://study.com/learn/lesson/terminal-velocity-formula-examples.html
- http://hyperphysics.phy-astr.gsu.edu/hbase/airfri2.html
- https://sciencing.com/calculate-terminal-velocity-6134922.html
- https://www.studysmarter.us/explanations/physics/mechanics-and-materials/terminal-velocity/