Terminal Velocity Calculator

Understanding Terminal Velocity

Terminal velocity is the constant speed that a freely falling object eventually reaches when the resistance of the medium through which it is falling prevents further acceleration. This phenomenon occurs due to the balance between the downward force of gravity and the upward resistive forces like drag or air resistance.

The Physics Behind Terminal Velocity

When an object falls through air, two primary forces act upon it:

• Weight force (Fg): The downward force due to gravity, calculated as Fg = mg, where m is mass and g is acceleration due to gravity (9.81 m/s²)
• Drag force (Fd): The upward resistive force that depends on the object's speed, calculated as Fd = ½ρACdv², where ρ is air density, A is cross-sectional area, Cd is drag coefficient, and v is velocity

Initially, as an object begins to fall, its weight exceeds the drag force, causing acceleration. However, as velocity increases, the drag force also increases until it equals the weight force. At this point, the net force becomes zero, and the object continues to fall at a constant velocity—the terminal velocity.

Factors Affecting Terminal Velocity

Several key factors influence an object's terminal velocity:

1. Mass and Weight

Heavier objects generally have higher terminal velocities. As mass increases, the weight force increases proportionally, requiring a greater drag force (and thus higher velocity) to achieve equilibrium.

2. Cross-sectional Area

The cross-sectional area perpendicular to the direction of motion significantly affects terminal velocity. A larger area results in more air resistance and a lower terminal velocity. This explains why a skydiver can reduce their falling speed by spreading their arms and legs to increase their effective area.

3. Drag Coefficient

The drag coefficient represents the aerodynamic efficiency of an object's shape. Objects with streamlined shapes (low drag coefficients) experience less air resistance and thus higher terminal velocities compared to objects with irregular shapes and high drag coefficients.

4. Fluid Density

Terminal velocity is inversely proportional to the square root of fluid density. In denser fluids (like water compared to air), objects reach their terminal velocity more quickly and the terminal velocity is lower. This explains why objects fall more slowly in water than in air.

Terminal Velocity in Different Scenarios

Skydiving

A typical skydiver in a belly-to-earth position (maximizing air resistance) has a terminal velocity of about 195 km/h (54 m/s). By changing body position to a head-down dive (minimizing air resistance), the same skydiver can reach speeds up to 320 km/h (90 m/s).

Small Objects and Stokes' Law

For very small objects like dust particles or small droplets, the drag force is proportional to velocity rather than velocity squared. This relationship is described by Stokes' Law:

Where η is the fluid viscosity, r is the radius of the particle, and v is the velocity. This results in much lower terminal velocities for tiny objects, explaining why dust particles can remain suspended in air for long periods.

Rain

Raindrops typically reach terminal velocities between 2 m/s for small drizzle drops to 9 m/s for large raindrops. Their terminal velocity is limited by their size and tendency to deform or break apart at higher speeds due to increasing air resistance.

Applications and Implications

Understanding terminal velocity has numerous practical applications:

• Design of parachutes and air brakes
• Development of aerodynamic vehicles
• Meteorology and precipitation analysis
• Safety engineering for falling objects
• Sedimentation processes in geology and chemistry

For engineering purposes, calculating the terminal velocity of objects is crucial in designing safety equipment, predicting behavior of falling objects, and optimizing the aerodynamics of vehicles and sporting equipment.

Terminal velocity is the maximum velocity attainable by an object as it falls through a fluid (air in this case).

To calculate terminal velocity, follow these steps:

1. 1
   Measure the mass of the object
2. 2
   Determine the cross-sectional area
3. 3
   Find the drag coefficient for the object's shape
4. 4
   Use the formula to calculate terminal velocity

Common drag coefficients for different shapes:

• Sphere: 0.47
• Circular flat plate: 1.17
• Streamlined body: 0.04
• Cube: 1.05