Lesson 7: Special Relativity | Modern Physics

Lesson at a Glance | He Tirohanga Whakamua

Do NowIf a spaceship at 0.8c shines a headlight, what speed does an Earth observer measure?10 min

Lorentz Factor & PostulatesSpeed of light constancy & calculating

γ = \frac{1}{\sqrt{1 - v^{2} / c^{2}}}

15 min

Cosmic Muon ParadoxResolving atmospheric muon detection via time dilation vs length contraction15 min

Portfolio EntryWrite Section 7: Special Relativity Kinematics & Lorentz Transformations10 min

Exit DrillCalculate Lorentz factor

γ

for a probe travelling at

0.80 c

(

γ = 1.67

)5 min

Ngā Whāinga Ako | Learning Intentions

Students will know

Einstein's 2 Postulates of Special Relativity (Physics laws identical in all inertial frames; $c$ is constant for all observers).
Time Dilation ( $t = γ t_{0}$ ): Moving clocks run slower relative to a stationary observer ( $t_{0}$ : proper time).
Length Contraction ( $L = \frac{L_{0}}{γ}$ ): Moving objects contract along their direction of motion ( $L_{0}$ : proper length).

Students will demonstrate

By resolving the Cosmic Ray Muon Paradox from both Earth and Muon reference frames.
By completing Section 7 of their Level 3 Modern Physics Mastery Portfolio.

Do Now | Tīmatanga Whakaaro (10 min)

Speed of Light Constancy Prompt:

"If you run at 15 km/h on a train moving at 80 km/h, a bystander sees you moving at 95 km/h. If a starship travelling at 0.8c turns on a laser beam, does an Earth observer measure the light travelling at 1.8c?"

Unpack: NO! Einstein's 2nd Postulate dictates that light ALWAYS travels at $c = 3.00 \times 10^{8} {ms}^{- 1}$ in a vacuum for ALL observers, regardless of motion. To keep $c$ constant, space and time themselves must flex ( $γ = \frac{1}{\sqrt{1 - v^{2} / c^{2}}}$ )!

Time Dilation & Cosmic Ray Muon Paradox (15 min)

1. Time Dilation ( $t = γ t_{0}$ )

Proper time ( $t_{0}$ ) is measured in the frame where the clock is at rest. Observed time ( $t$ ) is dilated (longer).

2. Length Contraction ( $L = \frac{L_{0}}{γ}$ )

Proper length ( $L_{0}$ ) is measured in the frame where the object is at rest. Observed length ( $L$ ) contracts in direction of motion.

📁 Physics Modern Portfolio — Section 7: Special Relativity Kinematics

Students open their Level 3 Physics Portfolio and complete Section 7:

Section 7 Requirements:

1. Lorentz Factor Graph: Plot $γ$ vs $v / c$ from $0$ to $0.99 c$ , marking $γ$ values at $0.6 c$ ( $γ = 1.25$ ), $0.8 c$ ( $γ = 1.67$ ), and $0.99 c$ ( $γ = 7.09$ ).

2. Muon Paradox Dual-Frame Proof: Show mathematically why a cosmic ray muon created $10 km$ high ( $v = 0.998 c, γ = 15.8, t_{1 / 2} = 2.2 μ s$ ) reaches Earth's surface in both Earth reference frame ( $t_{d i l a t i o n}$ ) and Muon reference frame ( $L_{c o n t r a c t i o n}$ ).

3. Excellence Twin Paradox Rationale: 1-paragraph explanation of why the travelling twin ages less than the stay-at-home Earth twin because turnaround acceleration breaks inertial symmetry.

Exit Verification | Ka Mutu Hoki (5 min)

Exit Check:

"My Section 7 calculates Lorentz factor gamma = 1.67 for 0.8c, proving time dilates to 1.67t0 and length contracts to L0/1.67."

Teacher Planning & NCEA Alignment

NCEA Level 3 Physics Alignment (3 Credits Internal):

Special Relativity: Demonstrate understanding of Einstein's postulates, inertial frames, Lorentz factor ( $γ$ ), proper time ( $t_{0}$ ), proper length ( $L_{0}$ ), time dilation, and length contraction.
Relativistic Evidence: Resolve the cosmic ray muon detection paradox.

Vocabulary: Special relativity, postulates, inertial frame, Lorentz factor ( $γ$ ), proper time ( $t_{0}$ ), proper length ( $L_{0}$ ), time dilation, length contraction, cosmic muon.

Other teaching approach: Guided Viewing & Problem Practice →

Lesson 7: Special Relativity — Kinematics & Space-Time