Learn7408Paper 3B · Option: Turning points in physicsSpecial relativityTime dilation and length contraction

Special relativity · Subtopic 1 of 1

Special Relativity

Einstein's postulates: the laws of physics are the same in every inertial frame, and the speed of light in a vacuum is the same for all observers. Consequences: moving clocks run slow, $t=\gamma t_0$ , and moving lengths shrink, $l=\frac{l_0}\gamma$ , where $\gamma=\frac1{\sqrt{1-v^2/c^2}}$ . A body's total energy is $E=\gamma m_0c^2$ .

AQA 7408 Paper 3B≈ 45 minRevision 1 · updated 2026-10-27

Section 1 of 14

Learning objectives

Explain the Michelson–Morley null result and state Einstein's postulates.
Use time dilation with proper time.
Use length contraction.
Use relativistic mass and $E=mc^2$ .

Pause and think

Muons made $10$ km up live about $2.2$ μs. Even at nearly light speed they'd travel only about $660$ m. Yet lots reach the ground. How?

Section 2 of 14

Core idea

Start hereMuons that shouldn’t arrive

Muons made by cosmic rays 10 km up should decay after a few hundred metres, even at almost light speed. Yet detectors at sea level catch plenty. The only explanation is that time itself runs differently for fast-moving things.

The big idea

If the speed of light is the same for everyone, time and length can't be. $t=\gamma t_0$ , $l=\frac{l_0}\gamma$ , $\gamma=\frac1{\sqrt{1-v^2/c^2}}$ .

1. Time dilation

The light clock

$t_0=\frac{2L}c$
At rest: light bounces up and down between mirrors $L$ apart.
$\left(\frac{ct}2\right)^2=L^2+\left(\frac{vt}2\right)^2$
Moving: light travels a diagonal path, still at speed $c$ .
$t^2(c^2-v^2)=4L^2=c^2t_0^2$
Multiply by 4 and collect.
$t=\frac{t_0}{\sqrt{1-v^2/c^2}}$
Rearrange.

Moving clocks run slow.

Worked exampleMuon journey

$v=0.995c$ , $t_0=2.2$ μs.

Goal: Mean distance in Earth's frame.
Strategy: Dilate, then $d=vt$ .

$\gamma=\frac1{\sqrt{1-0.990}}=10.0$
Lorentz factor.
$t=22\ \mu\text{s}$
Dilated lifetime.
$d=0.995\times3.0\times10^8\times22\times10^{-6}=6.6\text{ km}$
Distance.

AnswerAbout

6.6

km instead of

0.66

km.

✓ CheckIn the muon's frame the atmosphere is contracted by

\gamma

: same outcome ✓.

2. Length and energy

From postulates to predictions

Drawing diagram…

Everything follows from the two postulates.

Worked exampleFast electron

Electron at $0.60c$ , rest energy $0.511$ MeV.

Goal: Total energy.
Strategy: $E=\gamma m_0c^2$ .

$\gamma=\frac1{\sqrt{1-0.36}}=1.25$
Lorentz factor.
$E=1.25\times0.511=0.639\text{ MeV}$
Multiply.

Answer

0.639

MeV (kinetic energy

0.128

MeV).

✓ Check

\frac12mv^2

would give

0.092

MeV; relativity gives more ✓.

Common misconceptionProper quantities Proper time is the shortest measured interval; proper length is the longest measured length.

Key takeawayMichelson–Morley No fringe shift as the apparatus rotated: no ether, so light's speed doesn't depend on Earth's motion.

Pause and think

Why can't a spaceship reach $c$ ?

Summary

Two postulates.
$t=\gamma t_0$ .
$l=\frac{l_0}\gamma$ .
$E=\gamma m_0c^2$ .

Inertial frame: Frame moving at constant velocity.
Proper time: Time measured by a clock present at both events.
Lorentz factor: $\gamma=\frac1{\sqrt{1-v^2/c^2}}$ .

HistoryEinstein’s miracle year. In 1905, working as a patent clerk, Einstein published special relativity, the photoelectric explanation and Brownian motion, and

E=mc^2

followed months later.

Fun factGPS needs relativity. GPS satellite clocks would drift by about

38

μs a day without relativistic corrections, putting your position off by about

10

km a day.

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