Non-uniform Motion

2026 Syllabus Objectives

  1. Show a qualitative understanding of frictional forces and viscous/drag forces including air resistance
  2. Describe and explain qualitatively the motion of objects in a uniform gravitational field with air resistance
  3. Understand that objects moving against a resistive force may reach a terminal (constant) velocity

What is Non-uniform Motion?

Non-uniform motion occurs when an object's velocity changes over time. This happens when there is a resultant (net) force acting on the object. In real-world situations, objects often experience resistive forces that oppose their motion, causing them to speed up, slow down, or eventually move at constant velocity.

Frictional Forces

Friction is a resistive force that acts between surfaces in contact when they try to slide past each other. It always opposes the direction of motion or potential motion.

There are two types of friction:

  • Static friction: Acts on objects that are not moving yet but have a force trying to move them
  • Kinetic friction: Acts on objects that are already moving (usually slightly weaker than maximum static friction)

When you push a heavy box across the floor:

  • At first, static friction resists your push
  • Once the box starts moving, kinetic friction takes over
  • The harder you push, the faster the box moves (until friction balances your pushing force)

Friction and Motion: The Car Example

For a car traveling on a road, the relationship between the driving force and frictional force determines whether the car speeds up, slows down, or maintains constant velocity:

When accelerating:

  • The driving force (from the engine) is greater than the frictional force (from air resistance and road friction)
  • The resultant force acts forward
  • The car speeds up

When traveling at constant velocity:

  • The driving force equals the frictional force
  • There is no resultant force
  • The motion doesn't change

When decelerating:

  • The driving force is less than the frictional force
  • The resultant force acts backward (opposing motion)
  • The car slows down

Example calculation: A car of mass 800 kg has a forward driving force of 3000 N and accelerates at 2.0 m/s². What is the frictional force?

Step 1: Calculate the resultant force using Newton's Second Law (F = ma): Resultant force = 800 × 2.0 = 1600 N

Step 2: The resultant force is the difference between driving force and frictional force: Resultant force = Driving force − Frictional force 1600 = 3000 − Frictional force Frictional force = 3000 − 1600 = 1400 N

Drag Forces (Viscous Drag and Air Resistance)

Drag forces are resistive forces that act on objects moving through fluids. A fluid is any substance that can flow - this includes liquids (like water, oil) and gases (like air).

Key characteristics of drag forces:

  • They always oppose the direction of motion
  • They increase as the speed of the object increases (this is the most important property)
  • They are greater in denser fluids (moving through water creates more drag than moving through air)
  • They depend on the shape and size of the object (streamlined shapes experience less drag)

Air resistance is a specific type of drag force that acts on objects moving through air. When an object moves through air, it collides with air particles. The faster it moves, the more collisions happen per second, so the drag force increases.

Why Drag Force Matters

Air resistance becomes particularly important at high speeds. This is why:

  • Racing cyclists lean forward and wear tight clothing - to reduce air resistance
  • Cars are designed with smooth, curved shapes - to minimize drag
  • Parachutes work - they have a large surface area that creates massive air resistance

The fundamental principle: When an object moves faster, the drag force acting on it becomes stronger. When it slows down, the drag force decreases.

Motion in a Uniform Gravitational Field with Air Resistance

A uniform gravitational field means gravity pulls with the same strength everywhere (like near Earth's surface where g ≈ 10 m/s²). When objects fall through air, two main forces act on them:

  1. Weight (W = mg) - pulls the object downward (always constant for a given object)
  2. Air resistance (R) - pushes upward against the motion (changes with speed)

The Stages of Falling Through Air

Let's consider what happens when you drop an object from a height:

At the start of the fall:

  • The object starts from rest (velocity = 0)
  • Air resistance is zero (because the object isn't moving yet)
  • Weight pulls downward with full force
  • Resultant force = Weight (pointing down)
  • The object accelerates downward at g (about 10 m/s²)

As the object speeds up:

  • Velocity increases, so air resistance increases
  • Weight stays the same
  • Resultant force = Weight − Air resistance (pointing down, but smaller than before)
  • Acceleration decreases (the object still speeds up, but less rapidly)

Eventually:

  • Air resistance increases until it equals the weight
  • Resultant force = Weight − Air resistance = 0
  • Acceleration = 0
  • The object continues falling at constant velocity

This constant velocity is called terminal velocity.

Important note: If there were no air resistance (like on the Moon), the object would keep accelerating at g throughout its entire fall. Air resistance is what causes the acceleration to decrease and eventually become zero.

Terminal Velocity

Terminal velocity is the constant maximum velocity reached by an object moving through a fluid when the drag force equals the driving force (like weight for a falling object). At this point, the resultant force is zero, so there is no acceleration - the object moves at steady speed.

The Three Stages of Reaching Terminal Velocity

Stage 1 - Initial acceleration:

  • Object starts falling from rest
  • Weight > Air resistance
  • Large resultant force downward
  • Large acceleration downward (close to g)

Stage 2 - Decreasing acceleration:

  • Object's velocity increases
  • Air resistance increases (because drag increases with speed)
  • Weight > Air resistance (but the difference is smaller)
  • Resultant force downward (but smaller than before)
  • Acceleration downward (but decreasing)

Stage 3 - Terminal velocity reached:

  • Object continues to speed up until air resistance = weight
  • Weight = Air resistance
  • Resultant force = 0
  • Acceleration = 0
  • Object falls at constant velocity (terminal velocity)

Velocity-Time Graph for Terminal Velocity

The graph for an object reaching terminal velocity shows:

  • Starts at the origin (0,0)
  • Curve rises steeply at first (large acceleration)
  • Gradient gradually decreases (acceleration decreasing)
  • Finally levels off horizontally (acceleration = 0, terminal velocity reached)

The gradient of a velocity-time graph represents acceleration. As the gradient decreases, the acceleration decreases. When the line becomes horizontal, acceleration is zero.

Factors Affecting Terminal Velocity

1. Mass/Weight: Heavier objects have greater weight, so they need more air resistance to balance it. Therefore, heavier objects have higher terminal velocities.

2. Surface area: Larger surface area creates more air resistance at any given speed. Therefore, objects with larger surface area (like parachutes) have lower terminal velocities.

3. Shape: Streamlined objects experience less air resistance than irregular shapes, so they have higher terminal velocities.

Example: Two Skydivers

Consider two skydivers, A and B, where A has greater mass than B.

  • Skydiver A has more weight
  • For terminal velocity to be reached, air resistance must equal weight
  • Skydiver A needs more air resistance to balance their greater weight
  • Air resistance increases with speed
  • Therefore, Skydiver A must fall faster to generate enough air resistance
  • Skydiver A has a higher terminal velocity than Skydiver B

If they want to meet during the fall, Skydiver B should jump first. Skydiver A (falling faster) can then catch up.

Common misconception about parachutes: When a skydiver opens their parachute, they don't move upward - they decelerate to a new, lower terminal velocity. The parachute increases surface area dramatically, which increases air resistance. The air resistance becomes greater than weight temporarily, causing deceleration, until a new lower terminal velocity is reached.

Important Distinctions

When air resistance is negligible: If a question states that air resistance is "negligible," this means it is so small that it won't affect the motion. In this case, you can assume there are no drag forces acting on the object.

Upthrust vs. Drag:

  • Upthrust is an upward force caused by pressure differences in a fluid (it exists even when an object is stationary in a fluid)
  • Drag only exists when there is relative motion between the object and the fluid

For example, a stationary submarine underwater experiences upthrust but no drag. Once it starts moving, it experiences both upthrust and drag.

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