Terminal Velocity

What is Terminal Velocity?

Terminal velocity is the constant maximum speed reached by an object in free fall or falling through a fluid (such as air or water) when the force of air resistance or fluid resistance becomes equal in magnitude and opposite in direction to the force of gravity acting on the object.

At terminal velocity, the net force on the falling object becomes zero, and it no longer accelerates but continues to fall at a constant speed.

Explanation:

When an object falls freely under the influence of gravity, it experiences an acceleration due to gravity. However, as the object gains speed, it also experiences air resistance or fluid resistance, which acts in the opposite direction to its motion. Initially, the air resistance is relatively weak, and the object’s speed increases as it falls.

As the object’s speed increases, the air resistance also increases. Eventually, a point is reached where the air resistance becomes strong enough to balance the force of gravity. At this point, the net force acting on the object becomes zero, resulting in a constant velocity called terminal velocity.

For different objects, terminal velocity can vary based on their shape, mass, and surface area. Objects with larger surface areas and less mass experience higher air resistance and reach terminal velocity at lower speeds. Conversely, objects with smaller surface areas and more mass reach terminal velocity at higher speeds.

Terminal velocity is an essential concept in various fields, including physics, skydiving, and parachuting. When a skydiver jumps from an airplane, they initially accelerate due to gravity. As they gain speed, the air resistance increases until it balances out the force of gravity, and the skydiver reaches terminal velocity. At terminal velocity, the skydiver falls at a constant speed, without accelerating further.

In water, terminal velocity can be reached by objects falling through the fluid, like raindrops or objects dropped into water bodies. The concept of terminal velocity is also relevant in understanding the motion of objects falling through other fluids, such as gases or liquids, where fluid resistance plays a significant role in determining the object’s speed.

Laws Related to Terminal Velocity

Terminal velocity is governed by the laws of fluid dynamics, particularly the balance of forces acting on a falling object in a fluid medium (air or water). The main laws related to terminal velocity include:

1. Newton’s Second Law of Motion: This law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. In the context of terminal velocity, as an object falls, it accelerates due to the force of gravity. However, as the object’s speed increases, the force of air resistance (drag force) acting against its motion also increases. At terminal velocity, the net force becomes zero, resulting in a constant velocity with no further acceleration.
2. Drag Force Law: The drag force on a falling object is proportional to the square of its velocity through the fluid and is also influenced by the object’s shape and surface area. At low speeds, the drag force is relatively small, allowing the object to accelerate. As the object’s speed increases, the drag force becomes more significant, ultimately balancing out the force of gravity at terminal velocity.
3. Stokes’ Law (for Simple Spheres): Stokes’ Law is an approximation of drag force for small, smooth spheres falling in a fluid at low speeds. According to Stokes’ Law, the drag force (F_drag) on the sphere is directly proportional to its velocity (v) and the radius of the sphere (r). The equation is expressed as F_drag = 6πηrv, where η is the viscosity of the fluid.
4. Reynolds Number (Re): The Reynolds number is a dimensionless parameter used to predict the flow regime around an object in a fluid. For falling objects, it is used to determine whether the flow is laminar (smooth and ordered) or turbulent (chaotic and irregular). At low Reynolds numbers (Re < 1), the flow around the object is typically laminar, while at higher Reynolds numbers (Re > 2000), the flow is more likely to be turbulent. The type of flow can affect the object’s terminal velocity.

These laws work together to determine the terminal velocity of an object falling through a fluid. When the forces of gravity and drag reach equilibrium, the object’s acceleration becomes zero, and it falls at a constant speed known as terminal velocity. The specific terminal velocity of an object depends on factors such as its mass, shape, surface area, and the properties of the fluid (e.g., air density, water viscosity) it is falling through.

Expression for Terminal Velocity

The expression for terminal velocity using Stokes’ Law is given by: \[v_{\text{terminal}} = \frac{mg}{6\pi \eta r}\] where: \(v_{\text{terminal}}\) = Terminal velocity of the falling object (m/s)
\(m\) = Mass of the object (kg)
\(g\) = Acceleration due to gravity (approximately 9.81 m/s^2 on the Earth’s surface)
\(\eta\) = Viscosity of the fluid
\(r\) = Radius of the object

Derivation of Terminal Velocity

The force of gravity acting on the object is given by: \[F_{\text{gravity}} = mg\] The drag force (\(F_{\text{drag}}\)) is given by Stokes’ Law for a small, smooth sphere falling at low speeds in a fluid with low Reynolds numbers: \[F_{\text{drag}} = 6\pi \eta r v\] At terminal velocity, the object falls at a constant speed, which means there is no acceleration. The net force acting on the object is zero at terminal velocity, meaning the force of gravity is balanced by the drag force: \[F_{\text{gravity}} = F_{\text{drag}}\] Substituting the expressions for the forces, we get: \[mg = 6\pi \eta r v_{\text{terminal}}\] Now, we can solve for the terminal velocity (\(v_{\text{terminal}}\)): \[v_{\text{terminal}} = \frac{mg}{6\pi \eta r}\] This is the derived formula for the terminal velocity of an object falling through a fluid according to Stokes’ Law. It shows that the terminal velocity is directly proportional to the mass (\(m\)) of the object and inversely proportional to the viscosity of the fluid (\(\eta\)) and the radius (\(r\)) of the object. The larger the object’s mass or radius, or the smaller the fluid’s viscosity, the higher the terminal velocity will be.