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A.5.1 Galilean relativity (HL only)

Reference Frames: Inertial vs. Non-Inertial

In physics, reference frames are essential for describing the position and motion of objects.

Definition

Reference frame

A reference frame is a coordinate system that allows us to measure the position, velocity, and time of events.

Inertial Reference Frames

Definition

Inertial reference frame

An inertial reference frame is one where Newton’s first law holds true: an object at rest stays at rest, and an object in motion continues in a straight line at constant speed unless acted upon by a force.

Note

Inertial frames are either at rest or moving with constant velocity.

Non-Inertial Reference Frames

Definition

Non-inertial reference frame

A non-inertial reference frame is one that is accelerating.

In these frames, objects appear to experience fictitious forces, such as the sensation of being pushed back in a car that accelerates forward.

Note

Non-inertial frames require corrections for these fictitious forces to accurately describe motion.

Identifying Reference Frames

Consider these scenarios:

  • A train moving at constant speed on a straight track is an inertial frame.
  • A car accelerating around a curve is a non-inertial frame.

Self review

Which of the following are inertial frames?

  1. A satellite orbiting Earth
  2. A car moving at constant speed on a straight road
  3. A spinning merry-go-round

Galilean Transformations: Relating Two Reference Frames

  1. Galilean transformations provide a mathematical framework to relate the coordinates of an event in one inertial frame to another.
  2. These transformations assume that time is absolute and the same for all observers.

The Basic Equations

Consider two reference frames, S and S′:

  • S is stationary.
  • S′ moves with a constant velocity v relative to S along the x-axis.

The Galilean transformations are:

  1. Position Transformation: x′=x−vt
  2. Time Transformation: t′=t

Example

  1. Imagine a train moving at 10 m/s.
  2. A passenger drops a ball, and it hits the floor at x′ = 2 m and t′ = 1 s in the train’s frame (S′).
  3. For an observer on the ground (S), the ball’s position is: x=x′+vt=2 m+10 m/s×1 s=12 m
  4. The time remains the same: t=t′=1 s
Illustrating Galilean transformations.
Illustrating Galilean transformations.

Velocity Addition: Understanding Relative Velocity

Galilean transformations also help us understand how velocities are perceived differently in different frames.

The Galilean Velocity Addition Formula

If an object moves with velocity u′ in frame S′, its velocity u in frame S is given by:

u=u′+v

Example

  1. A train moves at 20 m/s relative to the ground.
  2. A passenger throws a ball forward at 5 m/s relative to the train.
  3. Velocity of the ball relative to the ground: u=u′+v=5 m/s+20 m/s=25 m/s

Self review

A boat moves at 4 m/s relative to a river flowing at 2 m/s. What is the boat’s velocity relative to the riverbank?

Limitations of Galilean Relativity

Galilean relativity works well for everyday speeds but breaks down at velocities approaching the speed of light.

Why Galilean Relativity Fails at High Speeds

  1. Galilean transformations assume that time is absolute and velocities add linearly.
  2. However, experiments show that the speed of light is constant for all observers, regardless of their motion.
This led to the development of Einstein’s theory of special relativity, which replaces Galilean transformations with Lorentz transformations.

Example

Imagine a spaceship moving at 0.9c (90% the speed of light) relative to Earth. It emits a light beam forward.

  • According to Galilean relativity, the light’s speed would be 1.9c for an Earth observer.
  • However, experiments confirm that the light’s speed remains c, not 1.9c.

Common Mistake

A common mistake is assuming that velocities always add linearly, even at high speeds.

This is only true for speeds much lower than the speed of light.

Reflection

  • Inertial frames move at a constant velocity; non-inertial frames accelerate.
  • Galilean transformations relate coordinates in different inertial frames, assuming time is absolute.
  • Velocity addition under Galilean relativity is linear: u=u′+v
  • Galilean relativity fails at high speeds, where the speed of light remains constant for all observers.

Theory of Knowledge

How did the discovery of the constancy of the speed of light challenge traditional notions of space and time? What does this tell us about the evolution of scientific knowledge?

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Questions

Recap questions

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Question 1

What are fictitious forces, and how do they affect the perception of motion in non-inertial reference frames?

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How do you calculate relative velocity using Galilean transformations?

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Note

Introduction to Reference Frames

A reference frame

is a fundamental concept in physics that provides a perspective from which we observe and measure events. It includes a coordinate system and a clock, allowing us to specify the position and time of events.

  • Reference frames are essential for describing motion because they provide a context for our observations.
  • Different observers may have different reference frames, leading to different measurements of the same event.

Analogy

Think of a reference frame like a camera viewpoint in a video game. Changing the camera position changes how you see the game world, but the game world itself doesn't change.

Example

When you watch a basketball game, you can describe the motion of the ball from different reference frames:

  • From the court's reference frame
  • From a moving camera's reference frame
  • From a player's reference frame

Definition

Reference frame

A coordinate system that allows us to measure the position, velocity, and time of events.

Common Mistake

Forgetting that all measurements of position and velocity are relative to a specific reference frame.