Terminal velocity is calculated by equating the force of gravity with the air drag force and solving for the velocity. The resulting formula is:
v = sqrt {(2 * m * g) / (d * A * C)}
Where:
- v = terminal velocity (m/s)
- m = mass of the object (kg)
- g = acceleration due to gravity (approximately 9.8 m/s²)
- d = density of the fluid (air in most cases) (kg/m³)
- A = projected area of the object (m²)
- C = drag coefficient (dimensionless)
Explanation of the Formula:
The formula stems from balancing the forces acting on a falling object: gravity and air resistance (drag).
- Force of Gravity (Fg): Fg = m * g. This force pulls the object downwards.
- Air Resistance (Fd): Fd = 0.5 * d * A * C * v². This force opposes the motion and increases with the square of the velocity.
At terminal velocity, Fg = Fd. Therefore:
m * g = 0.5 * d * A * C * v²
Solving for v yields the formula above.
Factors Affecting Terminal Velocity:
- Mass (m): A heavier object will generally have a higher terminal velocity, assuming other factors remain constant.
- Density of Fluid (d): An object falling through a denser fluid (like water) will have a lower terminal velocity than the same object falling through air.
- Projected Area (A): A larger projected area (the area the object presents to the fluid it's falling through) increases air resistance and decreases terminal velocity. Imagine a parachute versus a crumpled piece of paper.
- Drag Coefficient (C): This dimensionless coefficient depends on the shape of the object and how streamlined it is. A streamlined object has a lower drag coefficient than a blunt object. Values for common shapes can be found in engineering references.
Example:
Let's consider a skydiver:
- m = 75 kg
- g = 9.8 m/s²
- d = 1.225 kg/m³ (approximate density of air at sea level)
- A = 0.7 m² (approximate projected area of a skydiver in a belly-to-earth position)
- C = 1.2 (approximate drag coefficient for a skydiver)
v = sqrt {(2 * 75 kg * 9.8 m/s²) / (1.225 kg/m³ * 0.7 m² * 1.2)} ≈ 53.7 m/s (or about 120 mph)
Important Considerations:
- This calculation assumes a constant density of the fluid (air). In reality, air density decreases with altitude, which can affect terminal velocity during a long fall.
- The drag coefficient is often an approximation and can vary depending on the object's orientation and the flow of air around it.
- Wind can significantly affect the actual descent velocity.
In summary, calculating terminal velocity involves balancing the forces of gravity and air resistance and using the formula v = sqrt {(2 * m * g) / (d * A * C)}, understanding that factors like mass, density, projected area, and drag coefficient all play a significant role.
