What is terminal velocity?

Short Answer:

Terminal velocity is the constant maximum speed reached by a body when it falls freely through a fluid such as air or water. It occurs when the downward force of gravity is balanced by the upward force of air resistance or drag, making the net acceleration zero.

At terminal velocity, the body continues to fall at a uniform speed without any further increase in velocity. The value of terminal velocity depends on factors like the mass, shape, and surface area of the object, as well as the density and viscosity of the fluid through which it is moving.

Detailed Explanation:

Terminal Velocity

Terminal velocity is an important concept in fluid mechanics and motion through fluids. It refers to the steady speed attained by a freely falling body when the resistive forces of the fluid (such as air or water) exactly balance the gravitational pull on the object. At this point, the body experiences no net acceleration, and its velocity remains constant.

When a body starts falling through a fluid, it is initially accelerated due to gravity. However, as its speed increases, the drag force (air resistance) also increases. This drag acts in the opposite direction to the motion of the object, opposing its fall. Eventually, the drag force grows large enough to balance the weight of the object, resulting in zero net force and hence constant velocity. This constant speed is called the terminal velocity.

Forces Acting on a Falling Body

When an object falls through a fluid, three main forces act on it:

  1. Gravitational Force (Weight):
    This is the downward force acting due to gravity and is given by , where

    • m = mass of the body
    • g = acceleration due to gravity
  2. Buoyant Force:
    This is the upward force exerted by the displaced fluid. It depends on the fluid density and the volume of the displaced fluid.
  3. Drag Force (Air Resistance):
    This is the resistive force acting opposite to the motion of the body. It increases with velocity and depends on the shape, surface area, and nature of the fluid.

At terminal velocity:

Where  is the drag force and  is the buoyant force.

If the buoyant force is negligible (like for dense objects in air), then:

Mathematical Expression for Terminal Velocity

The drag force on an object can be expressed as:

Where,

  •  = Drag coefficient
  •  = Density of the fluid
  •  = Projected area of the body
  •  = Velocity of the body

At terminal velocity, the drag force equals the weight of the body (neglecting buoyancy):

Rearranging for terminal velocity :

This equation shows that terminal velocity increases with the mass of the object and decreases with higher drag coefficient, fluid density, or larger cross-sectional area.

Factors Affecting Terminal Velocity

  1. Mass of the Object:
    Heavier objects have higher terminal velocity because they experience greater gravitational force that overcomes air resistance more easily.
  2. Shape of the Object:
    Streamlined bodies like bullets or raindrops have low drag and hence higher terminal velocity. Irregular or flat objects have higher air resistance and lower terminal velocity.
  3. Surface Area:
    Larger surface area increases air resistance, reducing terminal velocity. For example, a parachute has a large surface area to create high drag and lower the terminal velocity.
  4. Density of the Fluid:
    A denser fluid offers more resistance, resulting in a lower terminal velocity. For instance, a stone falls slower in water than in air.
  5. Drag Coefficient (CD):
    The drag coefficient depends on the shape and smoothness of the object. A smaller drag coefficient indicates less resistance and higher terminal velocity.

Examples of Terminal Velocity

  1. Raindrops:
    When falling through the air, small raindrops reach a terminal velocity of about 9 m/s, while larger drops reach around 20 m/s. This is why raindrops don’t cause injury despite falling from clouds at great heights.
  2. Skydiver:
    A human skydiver reaches a terminal velocity of about 55 m/s (200 km/h) in a belly-down position. When they spread their limbs, they increase air resistance, reducing terminal velocity to around 50 m/s. In a head-first dive, terminal velocity can exceed 100 m/s due to a streamlined shape.
  3. Falling Objects in Water:
    A ball bearing dropped in oil or water reaches terminal velocity quickly because the viscous drag is much higher in liquids than in air.

Graphical Representation of Terminal Velocity

If velocity is plotted against time for a falling object, the curve starts with increasing slope (acceleration due to gravity) and gradually flattens as air resistance increases. The slope becomes zero at terminal velocity, indicating that acceleration has stopped and the object moves with constant speed.

Conclusion:

Terminal velocity is the final constant speed reached by a body falling through a fluid when the opposing drag force balances the gravitational force. It is an important concept for analyzing motion in fluids and depends on various factors such as mass, shape, surface area, and fluid properties. Understanding terminal velocity helps engineers and scientists in designing parachutes, vehicles, and other systems where fluid resistance plays a significant role in motion control and safety.