Year 12 Physics Module 7 ⏱ ~45 min 5 MC · 2 Short Answer Lesson 8 of 14

Special Relativity, Inertial Frames and Postulates

On 30 June 1905, Albert Einstein, then a patent clerk at the Swiss Patent Office in Bern, published "On the Electrodynamics of Moving Bodies," building on the 1887 Michelson–Morley null result (11 m optical path, expected 0.4 fringe shifts, measured less than 0.01). His two postulates, that physics is the same in all inertial frames, and that $c = 2.998 \times 10^8$ m/s is constant for all observers, overturned two centuries of Newtonian absolute space and time.

Today's hook: In 1887, Albert Michelson and Edward Morley at Case Western Reserve University, Cleveland, used an 11 m optical path interferometer and expected to detect Earth's motion through the aether as fringe shifts of 0.4. They measured less than 0.01. In June 1905 Albert Einstein at the Swiss Patent Office Bern used this null result to argue that $c$ must be the same for all observers, so if you shine a flashlight from a train moving at 100 km/h, the platform observer still measures the light at exactly $c$. How can that be?

0/5TASKS

Your guided path0 of 8 done · keep going

You’re here

Orient and predict

Predict how a torch beam behaves on a moving spaceship, then set your goals and vocabulary for special relativity.

Warm up first

Three quick questions from earlier lessons. Pulling old material back to mind before you learn something new makes the new material stick better, so this is not busywork.

Worksheets

Practise this lesson

Four printable worksheets that build from the foundations up to exam-style questions, start at whatever level suits you.

Build Foundations & guided practice Apply Application & data response Master Mastery tasks Exam Exam-style questions

Before you read, predict

You are on a train moving at 100 km/h. You shine a flashlight forward, and the light travels at $c$ relative to you.

According to classical (Galilean) physics, how fast does the light travel relative to a person standing on the platform?
If the speed of light is truly constant for all observers, what would the platform observer measure?
What must happen to time and space measurements if both observers measure the same speed of light?

Write your predictions before reading on, you will revisit them at the end.

Warm-up, in Galilean relativity, if you throw a ball at 10 m/s forward on a train moving at 20 m/s, a platform observer sees the ball at:

Learning Intentions

goals

Know, Inertial Frames

Frames moving at constant velocity
No preferred inertial frame exists
Galilean relativity vs Einsteinian relativity

Understand, Einstein's Postulates

Laws of physics are identical in all inertial frames
Speed of light in vacuum is constant for all observers
Michelson-Morley null result and its significance

Can Do, Apply Relativistic Concepts

Identify inertial vs non-inertial frames
Apply the postulates to simple scenarios
Calculate the Lorentz factor $\gamma$

Scan these before reading

vocab

Inertial frameA reference frame in which Newton's first law holds; a frame moving at constant velocity (no acceleration).

Special relativityEinstein's theory describing physics in inertial frames, based on the two postulates of relativity.

Luminiferous aetherA hypothetical light-carrying medium; the Michelson-Morley null result was inconsistent with the expected stationary-aether wind.

Lorentz factor$\gamma = \dfrac{1}{\sqrt{1 - v^2/c^2}}$; quantifies how much relativistic effects scale with velocity.

Null resultAn experimental outcome that detects no effect, often as significant as a positive result, as in Michelson-Morley.

Cross-lesson links: L02 introduced the Michelson–Morley 1887 null result as evidence that EM waves need no medium. L11 revisits the same experiment as the motivation for Einstein's June 1905 Swiss Patent Office Bern paper, the constancy of $c$ for all inertial observers. L12 applies the first concrete consequence: time dilation ($t = \gamma t_0$). L16 applies Einstein's other 1905 insight, the photon model of the photoelectric effect.

Core Content

Weigh the aether evidence and the two postulates

Follow Galilean relativity to the Michelson-Morley null result, then state Einstein’s two postulates.

From Galileo to Einstein

+5 XP

When common sense fails

Sit inside a smoothly gliding train with the blinds drawn. No experiment you can perform, dropping a ball, bouncing a laser off a mirror, measuring the weight of an object, can tell you whether the train is moving or stationary. Open the blinds and you can measure your speed relative to a tree or a platform, but there is no absolute "at rest." This is Galilean relativity: the laws of mechanics are the same in all inertial frames. Velocities simply add: throw a ball at 10 m/s forward on a train moving at 20 m/s and a platform observer measures the ball at 30 m/s.

But when James Clerk Maxwell derived his electromagnetic wave equations, they predicted a single speed for light: $c = 1/\sqrt{\mu_0 \varepsilon_0} \approx 3.00\times10^8$ m/s. This speed appeared without reference to any medium or observer, a profound departure from Galilean relativity.

Physicists assumed light travelled through an invisible medium called the luminiferous aether. If so, Earth's motion through the aether should create a detectable "aether wind." In 1887, Albert Michelson and Edward Morley performed an exquisitely precise interferometer experiment to measure this. Their result: no aether wind was detected. The speed of light was the same in all directions, regardless of Earth's motion.

Figure 1, Michelson-Morley interferometer: light split into two perpendicular paths and recombined. Earth's motion should have created a detectable path difference, but none was found (null result)

Einstein's solution was radical: discard the aether entirely. In his 1905 paper "On the Electrodynamics of Moving Bodies," he proposed two postulates:

The principle of relativity: The laws of physics are identical in all inertial frames of reference. No experiment can detect absolute motion.
The constancy of the speed of light: The speed of light in vacuum, $c$, is the same for all observers, regardless of their motion or the motion of the light source.

These postulates, seemingly simple, have extraordinary consequences: time dilation, length contraction, relativity of simultaneity, and the equivalence of mass and energy.

Stop & Check

A spaceship travels past Earth at 0.8$c$. The captain shines a laser forward. How fast does the laser light travel (a) relative to the captain, and (b) relative to an observer on Earth? Explain how this differs from Galilean velocity addition.

Galilean velocity addition conflicts with the invariant wave speed suggested by Maxwell's equations. Michelson and Morley's null result was inconsistent with the expected stationary-aether wind, within experimental sensitivity; it did not alone prove Einstein's later theory. Einstein's two postulates (1905) state that (1) the laws of physics are identical in all inertial frames and (2) every inertial observer measures the same vacuum light speed $c$. Relativistic effects follow from these postulates.

Write both postulates in full and note the Michelson-Morley significance.

The Michelson-Morley experiment was designed to detect:

Build the Lorentz factor and its domain

See how gamma behaves from everyday speeds up to a large fraction of $c$, and where it is defined.

The Lorentz Factor

+5 XP

The key to all relativistic effects

We just saw that Einstein's two postulates demand $c$ is constant for all observers. That raises a question: if $c$ never changes, what mathematical quantity tells us by how much time, length and energy change at high speeds? This card answers it → the Lorentz factor $\gamma$, which appears in every relativistic equation.

All relativistic effects are governed by the Lorentz factor $\gamma$ (gamma). It quantifies how much time dilation, length contraction, and energy increase occur at a given velocity.

Lorentz factor

$$\gamma = \dfrac{1}{\sqrt{1 - \dfrac{v^2}{c^2}}} = \dfrac{1}{\sqrt{1 - \beta^2}}$$

where $\beta = v/c$ is the velocity as a fraction of the speed of light.

At everyday speeds ($v \ll c$), $\gamma \approx 1$ and relativity reduces to classical physics. As $v$ approaches $c$, $\gamma$ grows without bound:

$v = 0.1c$: $\gamma = 1.005$ (0.5% effect)
$v = 0.5c$: $\gamma = 1.155$ (15.5% effect)
$v = 0.9c$: $\gamma = 2.29$ (129% effect)
$v = 0.99c$: $\gamma = 7.09$
$v = 0.999c$: $\gamma = 22.4$

Figure 2, The Lorentz factor $\gamma$ as a function of $v/c$. At low speeds $\gamma \approx 1$; it diverges as $v \to c$, making it impossible for a massive object to reach the speed of light

The Lorentz factor appears in time dilation, length contraction, relativistic momentum and energy equations; invariant mass does not increase with speed:

Key relativistic equations

$\Delta t = \gamma \,\Delta t_0$ (time dilation; $\Delta t_0$ = proper time)

$L = L_0/\gamma$ (length contraction; $L_0$ = proper length)

$E = \gamma mc^2$ (total relativistic energy)

$E_k = (\gamma-1)mc^2$ (relativistic kinetic energy)

HSC Tip, Calculator Discipline

Always calculate $\gamma$ first, then use it in subsequent equations. Work with $\beta = v/c$ to avoid large powers of 10. Never use classical KE $\frac{1}{2}mv^2$ for relativistic speeds, at $v = 0.98c$ the classical answer is off by nearly a factor of 10!

Stop & Check

Calculate $\gamma$ for a proton travelling at $0.95c$. If the proton's rest mass is $1.67\times10^{-27}$ kg, calculate its total energy and kinetic energy. ($c = 3.00\times10^8$ m/s)

Lorentz factor: $\gamma = 1/\sqrt{1-v^2/c^2}$; always $\geq 1$; $\to \infty$ as $v\to c$. Key values: $v=0.5c\Rightarrow\gamma=1.155$; $v=0.9c\Rightarrow\gamma=2.29$; $v=0.99c\Rightarrow\gamma=7.09$. It governs: $\Delta t = \gamma\Delta t_0$, $L = L_0/\gamma$, $E = \gamma mc^2$, $E_k = (\gamma-1)mc^2$, never use $\tfrac{1}{2}mv^2$ at relativistic speeds.

Record the formula and the four key relativistic equations that use $\gamma$.

A particle moves at $v = 0.6c$. Its Lorentz factor $\gamma$ is:

Solve Lorentz factor worked examples

Work gamma calculations step by step, including reading a speed back out of a given gamma.

Worked Example, Lorentz Factor Calculations

+5 XP

Getting comfortable with gamma

We just saw the Lorentz factor formula and the four equations it governs. That raises a question: how do you chain these equations together correctly in a multi-step exam problem involving time dilation, energy and potential difference? This card answers it → a four-step worked example at $v = 0.98c$.

Problem

An unstable muon is accelerated to 0.98$c$ in a particle accelerator.

Calculate the Lorentz factor $\gamma$.
If the muon's proper lifetime is $\tau_0 = 2.0\,\mu\text{s}$ for this worked example, calculate its dilated lifetime in the lab frame.
Calculate the electron's total energy and kinetic energy. ($m_e = 9.11\times10^{-31}$ kg)
Through what potential difference must the electron be accelerated to reach this speed?

Step 1, Lorentz factor

$\gamma = 1/\sqrt{1 - 0.98^2} = 1/\sqrt{1 - 0.9604} = 1/\sqrt{0.0396} = 1/0.199 = $ 5.03

Step 2, Time dilation

$\Delta t = \gamma \tau_0 = 5.03 \times 2.0 = $ 10.1 $\mu$s

Step 3, Energy

$E = \gamma m c^2 = 5.03 \times (9.11\times10^{-31})(3.00\times10^8)^2 = 5.03 \times 8.20\times10^{-14} = 4.12\times10^{-13}$ J

In eV: $E = 5.03 \times 511$ keV = 2.57 MeV

$E_k = (\gamma - 1)mc^2 = 4.03 \times 511$ keV = 2.06 MeV

Step 4, Potential difference

$E_k = qV$, so $V = E_k/e = 2.06\times10^6$ eV $/ e$ = 2.06 MV

Stop & Check

A proton travels at $0.87c$. Calculate $\gamma$, its total energy, and its kinetic energy in MeV. ($m_p c^2 = 938$ MeV)

Worked strategy at $v = 0.98c$: step 1, $\gamma = 5.03$. Step 2, time dilation: $\Delta t = 5.03\times 2.0\,\mu\text{s} = 10.1\,\mu\text{s}$. Step 3, total energy: $E = \gamma mc^2 = 2.57$ MeV; $E_k = (\gamma-1)mc^2 = 2.06$ MeV. Step 4, potential difference: $V = E_k/e = 2.06$ MV.

Write the four-step checklist: $\gamma$ first, then time dilation, energy, potential difference.

A muon travels at $0.99c$ ($\gamma \approx 7.09$). If its proper lifetime is 2.2 µs, its lifetime in Earth's frame is approximately:

Tell inertial from non-inertial frames

Decide which frames special relativity applies to, and what an accelerating observer detects.

Inertial vs Non-Inertial Frames

+5 XP

Recognising the difference matters for the HSC

We just saw how to calculate $\gamma$ and apply it in multi-step problems. That raises a question: special relativity only applies to inertial frames, so how do you identify whether a given frame qualifies? This card answers it → the definition of inertial vs non-inertial frames with HSC examples.

An inertial reference frame is one in which an object with no net force remains at rest or in uniform motion, Newton's first law holds. Any frame moving at constant velocity relative to an inertial frame is itself inertial.

A non-inertial frame is one that accelerates or rotates. Newton's first law does not hold without introducing fictitious forces (such as the centrifugal or Coriolis forces). Examples:

Inertial: a train moving at constant 200 km/h on a straight track; the surface of the Earth (approximately); a spacecraft coasting in deep space
Non-inertial: a car accelerating away from traffic lights; a spinning roundabout; a rocket thrusting through space

Figure 3, In an inertial frame, a ball remains stationary with no net force; in an accelerating (non-inertial) frame, a fictitious backward force appears to act on the ball

Special relativity applies only to inertial frames. General relativity (Einstein's later theory, 1915) extends this to include gravity and accelerating frames, but that is beyond the HSC scope.

Stop & Check

Classify each as inertial or non-inertial, with a reason: (a) ISS in circular orbit, (b) a car at constant speed on a curved road, (c) a spacecraft drifting at 0.5$c$ with engines off.

Inertial frame: constant velocity, Newton's 1st law holds, no fictitious forces. Non-inertial frame: accelerating or rotating, fictitious forces (centrifugal, Coriolis) appear. Special relativity applies only to inertial frames. A car cornering at constant speed is non-inertial (centripetal acceleration). A coasting spacecraft is inertial.

Copy the definition of inertial frame and three inertial/non-inertial examples.

An inertial frame must be at rest relative to Earth's surface.

Einstein's second postulate means that a platform observer and a train passenger both measure the same speed for a laser beam fired from the train.

The Michelson-Morley experiment provided evidence against the existence of the luminiferous aether.

Apply and consolidate

Build gamma fluency across a range of speeds, then evaluate what Michelson-Morley did and did not show.

Activity 1, Lorentz Factor Calculations

ApplyBand 4

Build fluency with $\gamma$ across the range of relativistic speeds

Calculate $\gamma$ for $v = 0.5c$, $0.9c$, and $0.99c$. Record your results and describe the trend.
At $v = 0.866c$, $\gamma = 2$. How much does a 1 m rod contract? How slow does a moving clock tick compared with a stationary one?
A muon is created at $0.99c$ with a proper lifetime of 2.2 µs. Calculate its lifetime in Earth's frame and the distance it travels before decaying.
Explain why $\gamma \to \infty$ as $v \to c$, and why this means massive objects cannot reach the speed of light.

Activity 2, Analyse the Michelson-Morley Experiment

UnderstandBand 5

Evaluate the experiment's role in building special relativity

Describe the apparatus and what Michelson and Morley expected to observe if the aether existed.
What did they actually observe? Why was this result surprising?
How did Einstein's postulates resolve the contradiction between classical mechanics and the Michelson-Morley result?
Is a null result in physics ever "unimportant"? Justify your answer using this experiment as an example.

Independent practice

Show what you can do without prompts

Answer a shuffled question-bank set, then write HSC-style responses on postulates, evidence and the Lorentz factor.

Quick recall, special relativity foundations

+5 XP

A fresh five-question set drawn from this lesson's bank, feedback shown immediately. +5 XP per correct · +25 XP all correct

Pick your answer, then rate your confidence, that tells the system what to drill next.

Short Answer, 7 marks

+5 XP

AnalyseBand 5(3 marks) 1. (a) Distinguish between an inertial and a non-inertial frame of reference, giving one example of each. (b) State Einstein's two postulates of special relativity. (c) Explain how the Michelson-Morley experiment contributed to the development of special relativity. (3 marks)

1 mark: correct distinction with examples · 1 mark: both postulates clearly stated · 1 mark: null result + how it motivated discarding the aether

EvaluateBand 6(4 marks) 2. A spacecraft moves at $0.95c$ relative to Earth. (a) Calculate its Lorentz factor. (b) Explain why $\gamma$ is dimensionless and why it can never be less than 1. (c) Predict what happens to $\gamma$ as $v$ approaches $c$. (d) Explain why substituting $v\geq c$ is outside the equation's physical domain for a massive object. (4 marks)

1 mark: correct $\gamma$ · 1 mark: dimensionless and lower bound · 1 mark: limiting behaviour · 1 mark: physical domain

ApplyBand 4(2 marks) 3. Classify each frame as inertial or non-inertial and justify your choice: (a) a spacecraft coasting in a straight line at constant velocity; (b) a car travelling at constant speed around a circular bend. (2 marks)

1 mark: coasting spacecraft is inertial · 1 mark: turning car is non-inertial because velocity direction changes

Show all answers

Multiple choice

MC answers and full explanations are shown inline as you complete each question. Use the retry button to attempt a fresh set drawn from the lesson bank.

Short Answer, Model Answers

Q1 (a): An inertial frame moves at constant velocity, Newton's first law holds with no fictitious forces (e.g. a train coasting at 200 km/h). A non-inertial frame accelerates or rotates, fictitious forces appear (e.g. a car braking). (1 mark)

Q1 (b): Postulate 1: The laws of physics are identical in all inertial frames, no experiment can detect absolute uniform motion. Postulate 2: The speed of light in vacuum is $c$ for all observers, regardless of the motion of the source or observer. (1 mark)

Q1 (c): Michelson and Morley observed no fringe shift at the size expected from Earth's motion through a stationary aether. Within the experiment's sensitivity, this null result was inconsistent with the expected aether wind. It undermined that aether model and helped motivate later relativistic accounts, but did not alone prove Einstein's second postulate. (1 mark)

Q2 (a): $\gamma = 1/\sqrt{1-0.95^2} = 1/\sqrt{0.0975} = $ 3.20. (1 mark)

Q2 (b): $v/c$ is a ratio of two speeds, so it and $\gamma$ are dimensionless. For a massive object $0\leq v^2/c^2<1$, making the square-root denominator at most 1 and therefore $\gamma\geq1$. (1 mark)

Q2 (c): As $v\to c$, $1-v^2/c^2\to0$ and $\gamma$ increases without bound. (1 mark)

Q2 (d): At $v=c$ the denominator is zero; above $c$ it is not real. A massive object cannot be accelerated to or through $c$, so those substitutions lie outside the model's physical domain for massive particles. (1 mark)

Q3 (a): The coasting spacecraft is an inertial frame because its velocity is constant and Newton's first law applies without fictitious forces. (1 mark)

Q3 (b): The car is non-inertial. Although its speed is constant, its direction and therefore its velocity change, so it has centripetal acceleration. (1 mark)

Final step

Retrieve, reflect and finish

Check what stuck, revisit your opening predictions and record the rule you will reuse.

Check what actually stuck

Take the full module quiz

quiz

A full module quiz covering every lesson in this module, not just this one. Set aside a decent block of time and treat it like a real assessment.

Start the module quiz →

How did your thinking change?

Look back at your Think First answers:

Did you predict that classical physics gives $c + 100\,\text{km/h}$ for light speed relative to the platform? That is Galilean velocity addition: $u' = u + v$.
Did you predict that both observers measure exactly $c$? This is Einstein's second postulate, and it defies classical intuition entirely.
Did you predict that time must slow down and lengths must contract to keep $c$ constant? Time dilation and length contraction are the direct consequences of the constancy of the speed of light.

The hook answer: the platform observer also measures the laser at $c$, not $c + 100\,\text{km/h}$. This forces time and space to be relative to the observer, not absolute.

The historical anchor: in 1887, Albert Michelson and Edward Morley at Case Western Reserve University, Cleveland, used an 11 m optical path interferometer and expected aether-drift fringe shifts of 0.4. They measured less than 0.01, a profound null result. In June 1905, Albert Einstein at the Swiss Patent Office, Bern, published "On the Electrodynamics of Moving Bodies," using this null result to argue that the speed of light is constant for all inertial observers. Einstein's two postulates, the principle of relativity and the constancy of $c$, are the foundation of everything in L12–L15.

I have completed this lesson

← Checkpoint 3 Next: Time Dilation →

Module overview · Physics · Module quiz