Introduction
3. Galilean Transformations
Learning outcomes
- I can describe Galilean transformations between moving reference frames.
- I can calculate transformed positions and velocities using Galilean transformations.
- I can explain the assumptions behind Galilean relativity.
- I can compare observations made in different inertial frames.
- I can recognize the limitations of Galilean transformations.
Introduction
Imagine standing beside a road as a train passes by at 20 m/s. A passenger inside the train throws a ball forward at 5 m/s relative to the train. How fast is the ball moving?
The answer depends on who is measuring it. To the passenger, the ball travels at 5 m/s. To someone standing beside the track, the ball moves at 25 m/s because it already has the speed of the train.
This simple idea is described by Galilean transformations, developed by the Italian scientist Galileo Galilei. These transformations explain how measurements of position and velocity change between observers moving at constant velocities relative to one another.
What Are Galilean Transformations?
Galilean transformations are equations that relate measurements made in different inertial reference frames moving at constant velocity relative to each other.
They allow us to compare measurements of:
- Position.
- Velocity.
- Time.
They work well for everyday speeds that are much smaller than the speed of light.
Two Reference Frames
Consider two reference frames:
- Frame S: Standing on the ground.
- Frame S′: Moving at a constant velocity v relative to the ground.
Both frames are inertial because neither is accelerating.
An object may have different measured positions and velocities in these two frames.
Figure 1. Two observers in different inertial reference frames measure the same motion differently.
Position Transformation
If Frame S′ moves with constant velocity v relative to Frame S, then the position measured in the two frames is related by:
where:
- x = position in Frame S
- x′ = position in Frame S′
- v = relative velocity between the frames
- t = time
This equation tells us how the measured position changes when switching reference frames.
Velocity Transformation
Velocities transform according to:
where:
- u = velocity measured in Frame S
- u′ = velocity measured in Frame S′
- v = velocity of Frame S′ relative to Frame S
This is often called velocity addition or velocity subtraction, depending on the direction of motion.
Figure 2. The measured velocity of an object depends on the observer's reference frame.
Time in Galilean Transformations
One important assumption is that:
This means:
- Time passes at the same rate for all observers.
- All observers agree on the timing of events.
This assumption works well for everyday situations but is not true at speeds close to the speed of light.
Galilean Relativity
Galilean relativity states:
The laws of mechanics are the same in all inertial reference frames.
This means:
- No inertial frame is "special."
- Experiments performed inside a smoothly moving vehicle give the same mechanical results as those performed at rest.
For example:
A passenger tossing a ball straight upward on a smoothly moving train observes the same motion as someone tossing a ball while standing still on the ground.
Comparing Observations
Different observers measure different values for:
- Position.
- Velocity.
However, they agree on:
- The laws of mechanics.
- Acceleration (provided both frames are inertial).
Example:
A train moves at 30 m/s.
A passenger walks forward at 2 m/s relative to the train.
Measurements:
- Passenger: 2 m/s
- Observer on the ground: 32 m/s
Both observations are correct.
Figure 3. Different inertial observers measure different velocities but agree on the laws of mechanics.
Worked Example 1
Question
A train moves at 15 m/s relative to the ground.
A passenger throws a ball forward at 8 m/s relative to the train.
What speed does an observer on the ground measure?
Solution
Using Galilean velocity addition:
Answer: The observer on the ground measures 23 m/s.
Worked Example 2
Question
A cyclist rides at 12 m/s relative to the road.
A car moves beside the cyclist at 10 m/s.
What speed does the driver measure for the cyclist?
Solution
Answer: The cyclist appears to move at 2 m/s relative to the car.
Assumptions Behind Galilean Transformations
Galilean transformations assume that:
- Both reference frames are inertial.
- Relative motion occurs at constant velocity.
- Time is the same for all observers.
- Space is absolute.
- Speeds are much smaller than the speed of light.
These assumptions make calculations simple and accurate for everyday situations.
Limitations of Galilean Transformations
Galilean transformations do not work when:
- Objects move at speeds close to the speed of light.
- Relativistic effects become significant.
At very high speeds:
- Time is no longer the same for all observers.
- Lengths can appear different.
- Velocities no longer add simply.
In these situations, physicists use Einstein's Special Theory of Relativity and the Lorentz transformations instead.
Figure 4. Galilean transformations work well for everyday speeds but must be replaced by special relativity at speeds close to the speed of light.
Everyday Applications
Galilean transformations are useful for describing:
- Walking inside moving trains.
- Aircraft flying through moving air.
- Boats travelling in rivers.
- Moving walkways.
- Conveyor belts.
- Sports involving moving players or vehicles.
In these situations, speeds are much smaller than the speed of light, so Galilean transformations provide accurate results.
Why Galilean Transformations Matter
Galilean transformations help scientists and engineers:
- Compare measurements made by different observers.
- Understand relative motion.
- Analyse moving vehicles.
- Predict motion in everyday situations.
They also provide the historical foundation for Einstein's theory of relativity.
Figure 5. Galilean transformations describe relative motion in many everyday situations involving moving reference frames.
Real-World Connection
Pilots must consider relative motion whenever they fly. An aircraft's airspeed is measured relative to the surrounding air, while its ground speed depends on both the aircraft's motion and the wind. If a plane flies at 250 km/h through still air, its ground speed is also 250 km/h. However, with a 40 km/h tailwind, the ground speed becomes 290 km/h. These everyday calculations are examples of Galilean velocity addition.
Did You Know?
Galileo developed his ideas about relativity more than 300 years before Einstein. He argued that passengers inside the cabin of a smoothly sailing ship could perform experiments—such as dropping objects or watching fish swim in a bowl—without being able to tell whether the ship was moving at a constant speed or standing still. This thought experiment became one of the foundations of modern physics.
Key Terms
Galilean relativity – The principle that the laws of mechanics are the same in all inertial reference frames.
Galilean transformation – Equations that relate position, velocity, and time measurements between inertial reference frames moving at constant velocity relative to one another.
Inertial reference frame – A reference frame at rest or moving with constant velocity.
Position transformation – The equation relating position measurements in different inertial reference frames.
Relative velocity – The velocity of one object or reference frame compared with another.
Velocity transformation – The equation relating velocity measurements in different inertial reference frames.
Key Takeaways
- Galilean transformations describe how position and velocity change between inertial reference frames moving at constant velocity.
- Different observers measure different positions and velocities, but both measurements are correct within their own reference frames.
- Galilean relativity states that the laws of mechanics are the same in all inertial reference frames.
- Galilean transformations assume that time is the same for all observers and that relative speeds are much smaller than the speed of light.
- They are widely used to analyse everyday situations involving moving vehicles and observers.
- At speeds approaching the speed of light, Galilean transformations are no longer accurate and must be replaced by Einstein's Special Theory of Relativity.