Representing Terminal Velocity

Definition of Terminal Velocity

Terminal velocity is the constant speed that a falling object eventually reaches when the forces acting on it are balanced. At this point, the downward force of weight (due to gravity) is exactly equal to the upward force of air resistance (drag). Because these forces balance out, the object no longer accelerates and continues to fall at a steady speed.

Forces Acting on Falling Objects

When an object falls through the air, two main forces act on it:

• Weight: The force due to gravity pulling the object downwards. It depends on the mass of the object and the gravitational field strength (usually $9.8 \text{ m/s}^2$ on Earth).
• Air resistance (drag): The force of air pushing upwards against the object, opposing its motion. This force increases as the object's speed increases.

At the start of the fall, weight is much greater than air resistance, so the object accelerates downwards. As the speed increases, air resistance grows until it equals the weight. At this point, the resultant force is zero, so acceleration stops and the object falls at terminal velocity.

For example, a skydiver jumping from a plane initially accelerates downwards because gravity is stronger than air resistance. As they speed up, air resistance increases until it balances their weight, and they fall at a constant terminal velocity.

Graphical Representation

Terminal velocity can be represented on different types of graphs:

Velocity-Time Graph

This graph shows how the velocity of a falling object changes over time:

• At the start, the velocity increases steadily as the object accelerates.
• As air resistance builds, acceleration decreases.
• The graph levels off to a flat, horizontal line when terminal velocity is reached, showing constant velocity.

Force-Speed Graph

This graph shows how the forces acting on the object change with speed:

• Weight: Constant and does not change with speed (horizontal line).
• Drag (air resistance): Increases with speed, starting at zero and rising until it equals the weight.
• The point where the drag line meets the weight line corresponds to terminal velocity.

These graphs help visualise how forces and velocity change during a fall and how terminal velocity is reached.

Factors Affecting Terminal Velocity

Several factors influence the terminal velocity of a falling object:

• Shape and surface area: Objects with a larger surface area experience more air resistance, so they have a lower terminal velocity. For example, a feather falls more slowly than a stone because it has a larger surface area relative to its mass.
• Mass of the object: Heavier objects have a greater weight, so they require a larger air resistance force to balance it. This usually means they reach a higher terminal velocity.
• Density of the fluid (air): Terminal velocity is affected by the density of the fluid the object is falling through. For example, in thicker air (higher density), air resistance is greater, so terminal velocity is lower.

Understanding these factors is important for predicting how different objects fall and reach terminal velocity.

Learning Example

A small ball with a mass of 0.2 kg is dropped. Its weight is calculated by:

$\text{Weight} = \text{mass} \times g = 0.2 \times 9.8 = 1.96 \text{ N}$

As it falls, air resistance increases with speed. When the air resistance reaches 1.96 N, the forces balance, and the ball reaches terminal velocity, falling at a constant speed.