Terminal Velocity (Air Resistance)

1. Definition

When an object falls through air, it experiences:

• weight, W = mg (downwards)
• air resistance (drag), F_d (upwards, opposite to the motion)

Terminal velocity is the constant velocity reached when the resultant force is zero:

F_d = W ⇒ a = 0

2. Key Ideas

• At the start: speed is low → drag is small → resultant force is downward → the object accelerates.
• As speed increases: drag increases → resultant force decreases → acceleration decreases.
• At terminal velocity: drag = weight → resultant force = 0 → constant velocity.
• A larger surface area (e.g. parachute) increases drag, giving a smaller terminal velocity.
• Terminal velocity depends on the object (shape, surface area, mass) and the fluid (air density) (qualitatively).

3. Detailed Explanations

A. Forces on a falling object

The two main forces are:

• weight W = mg (downwards, approximately constant near Earth)
• drag F_d (upwards, increases as speed increases)

The resultant force while falling is:

Taking downwards as positive:

Fᵣₑₛᵤₗₜₐₙₜ = W - F_d

B. Why acceleration decreases

From Newton’s second law:

Fᵣₑₛᵤₗₜₐₙₜ = ma

At the start, F_d is small, so Fᵣₑₛᵤₗₜₐₙₜ is large and the object accelerates quickly.

As speed increases, F_d increases, so W-F_d becomes smaller. This makes the acceleration smaller.

C. Terminal velocity (balanced forces)

Eventually the drag becomes large enough that:

F_d = W

So the resultant force is zero. When Fᵣₑₛᵤₗₜₐₙₜ = 0, acceleration is zero and the object continues at constant velocity (terminal velocity).

See also: Balanced Forces and Newton’s First Law and Unbalanced Force.

D. Velocity–time graph (typical shape)

The graph below is schematic (shape only).

Falling with air resistance (schematic)

Schematic velocity–time graph for a falling object with air resistance, approaching a constant terminal velocity.

Scroll across the graph to read all labels.

Open full-size graph

View figure data

Values for Falling with air resistance (schematic)

Time (arbitrary units)	v–t
0	0
1	12
2	22
3	30
4	36
5	41
6	45
7	48
8	50
9	50
10	50

Values for Falling with air resistance (schematic)

Time (arbitrary units)	v–t
0	0
1	12
2	22
3	30
4	36
5	41
6	45
7	48
8	50
9	50
10	50

Key links to remember:

• gradient of a v–t graph is acceleration
• “flattening” means acceleration is decreasing
• horizontal line means acceleration is zero (terminal velocity)

See: Reading Kinematics Graphs.

E. Parachutes (two terminal velocities)

When a parachute opens, surface area increases, so drag increases suddenly:

• drag can become greater than weight, so the resultant force becomes upwards
• the skydiver is still moving downwards, so this upward resultant force causes a deceleration
• a new, smaller terminal velocity is reached when drag again equals weight

4. Common Mistakes

• Saying “acceleration is always g” even when air resistance is present.
• Saying “terminal velocity means zero velocity” (it means zero acceleration).
• Forgetting that terminal velocity occurs when forces are balanced, not when there are no forces.
• Mixing up the forces: weight acts downwards; drag acts upwards while falling.

5. Exam Tips

• Always start with a quick force statement: “weight down, drag up”.
• Use: Fᵣₑₛᵤₗₜₐₙₜ = W-F_d (for downward motion, taking downward as positive).
• If the question says “constant velocity” or “terminal velocity”, write:
  • resultant force = 0 → F_d = W
• For parachute questions, include the key idea: “drag increases because surface area increases”.

6. Worked Examples

7. Mind Stretchers