Terminal velocity and Newton's laws of motion

What you'll learn

This guide covers terminal velocity and Newton's three laws of motion as required by the AQA GCSE Physics specification. You'll learn how to apply Newton's laws to analyse motion, calculate forces, and explain why falling objects reach a maximum speed. These concepts form the foundation of mechanics and appear regularly in both Paper 1 and synoptic questions.

Key terms and definitions

Resultant force — the single force that has the same effect as all the individual forces acting on an object combined

Inertia — the tendency of an object to remain at rest or continue moving at constant velocity unless acted upon by a resultant force

Terminal velocity — the maximum constant velocity reached by a falling object when the upward resistive forces equal the downward force of weight

Drag force — a frictional force that opposes the motion of an object moving through a fluid (liquid or gas), also called air resistance in gases

Acceleration — the rate of change of velocity, measured in metres per second squared (m/s²)

Newton — the SI unit of force; one newton causes a 1 kg mass to accelerate at 1 m/s²

Equilibrium — the state when all forces acting on an object are balanced, resulting in zero resultant force

Core concepts

Newton's First Law of Motion

Newton's First Law states that an object will remain at rest or continue to move at a constant velocity unless acted upon by a resultant force. This law describes inertia — the resistance of objects to changes in their motion.

Key points about the First Law:

• If the resultant force on an object is zero, the object maintains constant velocity (which may be zero)
• A moving object doesn't need a force to keep it moving, only to change its motion
• The greater an object's mass, the greater its inertia

Common applications:

• A book resting on a table remains stationary because the upward contact force from the table equals the downward weight
• A football rolling on grass gradually slows because friction provides a resultant force opposing motion
• Passengers lurch forward when a bus brakes suddenly because their bodies tend to maintain constant velocity

Newton's Second Law of Motion

Newton's Second Law establishes the relationship between force, mass and acceleration:

F = ma

where:

• F = resultant force in newtons (N)
• m = mass in kilograms (kg)
• a = acceleration in metres per second squared (m/s²)

This equation tells us:

• The acceleration of an object is directly proportional to the resultant force acting on it
• The acceleration is inversely proportional to the object's mass
• The direction of acceleration is the same as the direction of the resultant force

Rearranging the equation:

• To find mass: m = F/a
• To find acceleration: a = F/m

A larger resultant force produces greater acceleration. A larger mass produces smaller acceleration for the same force. This explains why a car accelerates more slowly when fully loaded with passengers and luggage.

Newton's Third Law of Motion

Newton's Third Law states that whenever two objects interact, they exert equal and opposite forces on each other. These forces are often called action-reaction pairs.

Characteristics of action-reaction pairs:

• The forces are always equal in magnitude
• The forces are always opposite in direction
• The forces act on different objects (crucial for identification)
• The forces are always the same type (both gravitational, both contact forces, etc.)

Examples of action-reaction pairs:

• Earth pulls down on a ball (weight); the ball pulls up on Earth with an equal force
• A swimmer pushes water backwards; water pushes the swimmer forwards
• Rocket exhaust gases are pushed downwards; gases push the rocket upwards

Common misconception: Forces being balanced (Newton's First Law) is different from action-reaction pairs (Newton's Third Law). Balanced forces act on the same object; action-reaction forces act on different objects.

Forces and motion

When analysing motion problems at GCSE level, follow this process:

1. Identify all forces acting on the object
2. Determine the resultant force (may require vector addition)
3. Apply F = ma if the resultant force is non-zero
4. Consider whether forces are balanced (constant velocity) or unbalanced (acceleration)

For vertical motion:

• Objects accelerate downwards at approximately 9.8 m/s² near Earth's surface when air resistance is negligible
• Weight (W) = mass (m) × gravitational field strength (g), where g = 9.8 N/kg on Earth

For horizontal motion:

• Friction and drag oppose motion
• Driving forces (thrust, push, pull) cause acceleration in the direction applied

Terminal velocity in fluids

When an object falls through a fluid (air or liquid), it experiences two main forces:

• Weight (W) — acts downwards, remains constant
• Drag force (air resistance or water resistance) — acts upwards, increases with speed

The process of reaching terminal velocity:

Stage 1: Initial acceleration

• Object begins to fall from rest
• Weight > drag (small upward force at low speed)
• Resultant force downwards
• Object accelerates downwards

Stage 2: Decreasing acceleration

• As velocity increases, drag force increases (approximately proportional to v² at higher speeds)
• Weight remains constant
• Resultant force downwards decreases
• Acceleration decreases but velocity still increases

Stage 3: Terminal velocity

• Drag force increases until it equals weight
• Weight = drag (forces balanced)
• Resultant force = 0 N
• Acceleration = 0 m/s²
• Velocity remains constant at maximum value (terminal velocity)

Factors affecting terminal velocity:

• Mass/weight — heavier objects have higher terminal velocities
• Surface area — larger surface area means greater drag, lower terminal velocity
• Shape — streamlined shapes reduce drag, increasing terminal velocity
• Fluid density — denser fluids create more drag, reducing terminal velocity

Practical examples:

• Skydivers reach terminal velocity of approximately 55 m/s (200 km/h) in spread-eagle position
• Opening a parachute dramatically increases surface area, reducing terminal velocity to approximately 5 m/s for safe landing
• Raindrops reach terminal velocities of 2-9 m/s depending on size

Velocity-time graphs for terminal velocity

A velocity-time graph for an object reaching terminal velocity shows:

• Initially: steep positive gradient (large acceleration)
• Middle section: decreasing gradient (acceleration decreasing as drag increases)
• Final section: horizontal line (zero acceleration, constant velocity)

The gradient at any point gives the instantaneous acceleration. The area under the graph gives the distance travelled.

When a parachute opens:

• Sudden large increase in drag force
• Drag > weight
• Resultant force upwards (opposing motion)
• Object decelerates (negative acceleration but still moving downwards)
• Velocity decreases until drag again equals weight
• New, lower terminal velocity reached

Worked examples

Example 1: Newton's Second Law calculation

Question: A car of mass 1200 kg accelerates from rest. The engine provides a driving force of 3600 N. Friction and air resistance provide a total resistive force of 800 N. Calculate the acceleration of the car. [3 marks]

Solution:

Step 1: Calculate the resultant force Resultant force = driving force - resistive force Resultant force = 3600 - 800 = 2800 N [1 mark]

Step 2: Apply Newton's Second Law (F = ma) 2800 = 1200 × a [1 mark]

Step 3: Rearrange to find acceleration a = 2800 ÷ 1200 = 2.33 m/s² (or 2.3 m/s² to 2 s.f.) [1 mark]

Example 2: Terminal velocity explanation

Question: A skydiver jumps from an aircraft. Explain why the skydiver initially accelerates but then reaches a constant velocity. [4 marks]

Model answer:

Initially, the only force acting is the weight of the skydiver pulling them downwards [1 mark]. As the skydiver falls, air resistance acts upwards and increases as speed increases [1 mark]. The resultant downward force decreases, so acceleration decreases [1 mark]. Eventually air resistance equals weight, giving zero resultant force, so the skydiver stops accelerating and moves at constant velocity (terminal velocity) [1 mark].

Example 3: Newton's Third Law identification

Question: A book rests on a table. State one action-reaction pair of forces in this situation. [2 marks]

Model answer:

The book pushes down on the table [1 mark] and the table pushes up on the book [1 mark].

Alternative answer: The Earth pulls down on the book (weight) [1 mark] and the book pulls up on the Earth [1 mark].

Note: "Weight of book downwards" and "contact force from table upwards" would score 0 marks — these are balanced forces acting on the same object, not an action-reaction pair.

Common mistakes and how to avoid them

• Confusing balanced forces with action-reaction pairs — Balanced forces act on the same object and can have different types (e.g., weight and contact force). Action-reaction pairs act on different objects and are always the same type of force.

• Thinking objects need a force to keep moving — Newton's First Law tells us objects maintain constant velocity without a resultant force. Only changing motion requires a resultant force.

• Stating terminal velocity occurs when forces are "equal" — Be specific: terminal velocity occurs when upward drag force equals downward weight, giving zero resultant force and zero acceleration.

• Misapplying F = ma when forces are balanced — If velocity is constant (including zero), acceleration is zero, so resultant force must be zero. Don't use individual forces in the equation; only use the resultant force.

• Forgetting units in calculations — Always include units: force in N, mass in kg, acceleration in m/s². Marks are often lost for missing units in final answers.

• Describing terminal velocity as "when forces balance" — While correct, better answers specify which forces: "when drag force equals weight" or "when air resistance equals weight, giving zero resultant force."

Exam technique for "Terminal velocity and Newton's laws of motion"

• Command word "explain" — You must give reasons using physics principles. For terminal velocity questions, describe how forces change and link to changes in acceleration. Expect 3-4 marks for full explanations requiring multiple linked points.

• Using F = ma — Show all working: identify resultant force, substitute values with units, rearrange if needed, calculate and state the unit. Even if your answer is wrong, you can gain method marks (typically 2-3 marks for calculation questions).

• Velocity-time graph questions — Remember the gradient gives acceleration and the area gives distance. For terminal velocity, describe how the gradient changes (steep, then decreasing, then zero).

• Identifying action-reaction pairs — Both forces must act on different objects and be the same type. State clearly what exerts the force, the direction, and what object it acts on. Worth typically 2 marks (1 for each force).

Quick revision summary

Newton's First Law: objects maintain constant velocity unless acted on by a resultant force. Newton's Second Law: F = ma links force, mass and acceleration. Newton's Third Law: interactions produce equal and opposite forces on different objects. Terminal velocity occurs when drag force equals weight during free fall, giving zero resultant force and constant velocity. Drag increases with speed until equilibrium is reached. Heavier or streamlined objects have higher terminal velocities. These principles explain motion in everyday contexts from vehicles to skydivers.