Lesson 8: Relativistic Energy | Modern Physics

Lesson at a Glance | He Tirohanga Whakamua

Do NowWhy no object with rest mass can ever reach or exceed the speed of light10 min

Relativistic Momentum & EnergySolving

p = γ m_{0} v

and

E_{k} = (γ - 1) m_{0} c^{2}

15 min

Particle Accelerator PhysicsCreating heavy particles from kinetic energy collisions at CERN15 min

Portfolio EntryWrite Section 8: Relativistic Dynamics & Mass-Energy Matrix10 min

Exit DrillCalculate kinetic energy of an electron travelling at

0.95 c

(

γ = 3.20

)5 min

Ngā Whāinga Ako | Learning Intentions

Students will know

Relativistic Momentum ( $p = γ m_{0} v$ ): Classical momentum ( $p = m v$ ) breaks down as velocity approaches $c$ . As $v \to c$ , $p \to \infty$ .
Relativistic Energy ( $E = γ m_{0} c^{2}$ ): Total energy consists of Rest Mass Energy ( $E_{0} = m_{0} c^{2}$ ) plus Relativistic Kinetic Energy ( $E_{k} = (γ - 1) m_{0} c^{2}$ ).
Why $c$ is the absolute cosmic speed limit: Accelerating a massive object to $c$ requires infinite kinetic energy ( ${lim}_{v \to c} γ = \infty$ ).

Students will demonstrate

By calculating total energy and kinetic energy for high-speed protons accelerated in synchrotrons.
By completing Section 8 of their Level 3 Modern Physics Mastery Portfolio.

Do Now | Tīmatanga Whakaaro (10 min)

Universal Speed Limit Prompt:

"Classical Newtonian physics says $F = m a$ , meaning a constant push force will accelerate an electron infinitely past the speed of light. Why does an electron in a particle accelerator approach $0.9999 c$ but never reach $1.0 c$ ?"

Unpack: Because momentum and energy become relativistic! $p = γ m_{0} v$ and $E = γ m_{0} c^{2}$ . As speed approaches $c$ , $γ \to \infty$ , meaning additional energy input goes into increasing the particle's relativistic energy and inertia rather than increasing its speed!

Relativistic Energy Equations (15 min)

1. Total Energy Equation

$E_{t o t a l} = γ m_{0} c^{2}$
Rest Energy: $E_{0} = m_{0} c^{2}$
Kinetic Energy: $E_{k} = (γ - 1) m_{0} c^{2}$ .

2. Momentum-Energy Relation

$E^{2} = (p c)^{2} + (m_{0} c^{2})^{2}$
For massless photons ( $m_{0} = 0$ ), $E = p c \to p = \frac{h}{λ}$ .

📁 Physics Modern Portfolio — Section 8: Relativistic Energy & Momentum

Students open their Level 3 Physics Portfolio and complete Section 8:

Section 8 Requirements:

1. Kinetic Energy Comparison Graph: Plot Classical $E_{k} = \frac{1}{2} m v^{2}$ vs Relativistic $E_{k} = (γ - 1) m_{0} c^{2}$ from $0$ to $0.99 c$ .

2. Proton Synchrotron Solver: Calculate rest energy $E_{0}$ , total energy $E$ , relativistic momentum $p$ , and kinetic energy $E_{k}$ in MeV for a proton ( $m_{p} = 1.67 \times 10^{- 27} kg$ ) moving at $0.95 c$ ( $γ = 3.20$ ).

3. Excellence Mass-Energy Creation Rationale: 1-paragraph explanation of how high-energy proton collisions at CERN transform kinetic energy into rest mass to create heavy particles ( $p + p \to p + p + H^{0}$ ).

Exit Verification | Ka Mutu Hoki (5 min)

Exit Check:

"My Section 8 calculates total energy E = 3.20 E0 and kinetic energy Ek = 2.20 E0 (2064 MeV) for a 0.95c proton, proving classical Ek underestimates energy."

Teacher Planning & NCEA Alignment

NCEA Level 3 Physics Alignment (3 Credits Internal):

Relativistic Energy & Momentum: Demonstrate understanding of relativistic momentum ( $p = γ m_{0} v$ ), rest mass energy ( $E_{0} = m_{0} c^{2}$ ), total energy ( $E = γ m_{0} c^{2}$ ), kinetic energy ( $E_{k} = (γ - 1) m_{0} c^{2}$ ), and mass creation in particle physics.

Vocabulary: Relativistic momentum, rest mass energy ( $E_{0}$ ), relativistic kinetic energy ( $E_{k}$ ), total energy ( $E$ ), cosmic speed limit ( $c$ ), synchrotron.

Other teaching approach: Guided Viewing & Problem Practice →

Lesson 8: Relativistic Mass, Momentum & Energy