Objects falling through air or any fluid experience forces that alter their acceleration and final speed. Understanding falling with drag connects gravitational motion, resistive forces, and terminal velocity behaviour.

Motion in a Uniform Gravitational Field

When an object falls freely under gravity, it accelerates downwards at approximately 9.81 m s⁻², assuming the field is uniform—that is, gravitational strength is constant throughout the region of motion. In the absence of air resistance, all objects would experience uniform acceleration, and their velocity–time graph would be a straight line increasing steadily with time.

However, in reality, objects fall through fluids (liquids or gases) that exert an opposing drag force. The inclusion of drag makes the acceleration non-uniform, meaning it decreases over time until a steady speed, the terminal velocity, is reached.

Forces Acting on a Falling Object

When considering vertical motion through a fluid, two main forces act:

Free-body diagrams for a falling skydiver: initially D = 0 and acceleration ≈ g; as speed rises, drag increases, reducing the net force; at terminal velocity, drag = weight and acceleration = 0. The diagram focuses only on the two forces required by the syllabus and avoids extraneous details (minor extra text on the page discusses dynamic equilibrium, consistent with terminal motion). Source

• Weight (W) — the downward gravitational force on the object, equal to its mass multiplied by the gravitational field strength.

• Drag (D) — the upward resistive force opposing motion through the fluid.

As the object falls:

1. Initially, weight is greater than drag, so the object accelerates downwards.

2. As speed increases, drag increases due to greater relative motion between object and fluid.

3. Eventually, drag grows until it equals weight.

4. At this point, the net force is zero, and the object continues at constant velocity — the terminal velocity.

This balance of forces determines the final motion of any falling object in a fluid.

Nature and Dependence of Drag

Drag depends on several interrelated factors:

• Speed of the object — drag typically increases with speed, often proportional to the square of the velocity at higher speeds.

• Shape and surface area — larger cross-sectional areas and irregular shapes experience greater drag.

• Nature of the fluid — more viscous fluids (like oil) exert higher drag than less viscous ones (like air).

At low speeds or in highly viscous fluids, drag may be approximately proportional to velocity (v). At higher speeds, especially in air, drag tends to be proportional to v², leading to different mathematical treatments of motion.

Changing Acceleration During the Fall

At the instant an object begins to fall, drag = 0, and the net force = weight, producing maximum acceleration equal to g. As the object speeds up:

• Drag increases, reducing the net force.

• Consequently, acceleration decreases.

• When drag = weight, acceleration = 0, and the object moves at a constant velocity.

The overall motion is thus non-uniform acceleration — rapid at first, gradually diminishing as the object approaches terminal velocity.

Describing the Velocity–Time Graph

The velocity–time graph for falling with drag has a distinctive curved shape:

Velocity–time graph for a skydiver falling with air resistance. The slope (acceleration) is large at first, then decreases as drag grows, and the velocity asymptotically approaches a constant terminal velocity. Labels are minimal and clear, matching OCR’s required qualitative sketch. Source

1. Initial Region – The graph starts at zero velocity with a steep slope, showing large acceleration.

2. Intermediate Region – The curve begins to flatten as acceleration decreases and drag grows.

3. Final Region – The graph levels off horizontally as the object reaches terminal velocity, where velocity is constant.

This contrasts with the straight-line graph of free fall without drag, which continues to increase linearly.

Students must be able to sketch and interpret this curve, identifying the point of terminal velocity and the changing gradient (acceleration) throughout the motion.

Relationship Between Forces and Velocity

EQUATION
—-----------------------------------------------------------------
Newton’s Second Law (F = ma)
F = ma
F = resultant force on the object (N)
m = mass of the object (kg)
a = acceleration (m s⁻²)
—-----------------------------------------------------------------

For a falling object:
Resultant force = Weight − Drag
Therefore: ma = mg − D

As D increases with velocity, a must decrease to maintain this equality. Eventually, when D = mg, a = 0, confirming terminal velocity.

This equation helps describe the continuous balance of forces and how acceleration reduces during descent.

Qualitative Motion Description

A typical sequence for an object falling with drag is:

• Release: Acceleration = g (maximum value).

• Early motion: Speed increases, drag begins to act but is small.

• Midway: Drag becomes significant; acceleration falls below g.

• Terminal phase: Drag = weight, net force = 0, and motion becomes uniform at terminal velocity.

The transition between these stages depends on the mass, cross-sectional area, and fluid properties. Denser or more streamlined objects achieve higher terminal velocities because they experience smaller drag relative to their weight.

Velocity–Time Behaviour in Context

The velocity–time behaviour summarises how forces and motion interrelate:

• The gradient of the curve gives acceleration, which steadily decreases.

• The area under the curve represents displacement, showing how distance fallen increases non-linearly.

• The horizontal asymptote of the graph corresponds to terminal velocity, beyond which velocity remains constant.

Understanding this relationship helps explain why, for example, raindrops, skydivers, or small beads in oil do not continue accelerating indefinitely but instead reach a steady fall speed determined by their drag and weight balance.

Practical Significance

The concept of falling with drag has real-world applications:

• Parachuting: Parachutes increase surface area dramatically, boosting drag and reducing terminal velocity for safe landings.

• Vehicle aerodynamics: Engineers design shapes to minimise drag and optimise speed.

• Sedimentation: Particles settling in liquids demonstrate similar behaviour, governed by the same principles.

By recognising the interplay between gravitational pull and drag resistance, physicists can model and predict how different objects fall in various environments — a vital understanding for both theoretical analysis and experimental investigation.