Physics • Year 12 • Module 7 • Lesson 14
Relativistic Mass, Momentum and Energy
Lock in the key equations, vocabulary, and concepts of relativistic energy before tackling harder problems.
1. Term–definition match
The definitions below are shuffled. Write the matching term from this list in the right-hand column: rest energy, relativistic kinetic energy, mass-energy equivalence, Lorentz factor, relativistic momentum, total relativistic energy, mass defect, particle-antiparticle annihilation. 8 marks (1 each)
| # | Definition | Matching term |
|---|---|---|
| 1.1 | The energy stored in a stationary particle due to its mass alone; $E_0 = mc^2$. | |
| 1.2 | The factor $\gamma = 1/\sqrt{1 - v^2/c^2}$ that quantifies how much time, length, and energy are altered at speed $v$. | |
| 1.3 | The principle that mass and energy are interchangeable forms of the same quantity, related by $E = mc^2$. | |
| 1.4 | The total energy of a moving particle: $E = \gamma mc^2$. | |
| 1.5 | The kinetic energy of a relativistic particle: $E_k = (\gamma - 1)mc^2$; diverges to infinity as $v \to c$. | |
| 1.6 | The relativistic generalisation of momentum: $p = \gamma mv$. | |
| 1.7 | The small difference between the total mass of reactants and products in a nuclear reaction; this "missing" mass is converted to energy. | |
| 1.8 | The process where a particle meets its corresponding antiparticle and their entire rest mass is converted into photons. |
2. True or false — with correction
Circle T or F. If the statement is false, write the corrected version on the line below it. 12 marks (1 T/F + 1 correction each)
2.1 Classical kinetic energy $E_k = \frac{1}{2}mv^2$ is accurate for all speeds up to the speed of light. T / F
2.2 A particle at rest has zero total energy because it has no kinetic energy. T / F
2.3 As a particle's speed approaches $c$, its kinetic energy approaches infinity. T / F
2.4 Massless particles such as photons travel at speeds less than $c$ because they have no mass to push against. T / F
2.5 The kinetic energy of a particle equals its total energy minus its rest energy. T / F
2.6 In a nuclear reaction, the mass defect is lost and cannot be accounted for by conservation of energy. T / F
3. Fill-in-the-blank paragraph
Use the word bank to complete the passage. Each word or phrase is used once. 8 marks (1 per blank)
Word bank:
gamma-ray photons · infinite · kinetic · mass-energy equivalence · massless · momentum · rest energy · Lorentz factor
The ___________ $\gamma$ grows without bound as a particle's speed approaches $c$, which means the ___________ energy $E_k = (\gamma - 1)mc^2$ also becomes ___________—the reason massive particles can never reach the speed of light. Even at rest, a particle possesses ___________ $E_0 = mc^2$. The principle of ___________ tells us that mass and energy are equivalent. Relativistic ___________ is given by $p = \gamma mv$, not the classical $mv$. Photons are ___________ particles and always travel at exactly $c$. When an electron meets a positron, they annihilate into two ___________.
4. Function recall
Answer each question in 1–2 sentences using precise terms from the lesson. 8 marks (2 each)
4.1 State the physical reason why no massive particle can ever be accelerated to exactly the speed of light $c$.
4.2 What is the difference between total relativistic energy and relativistic kinetic energy? State the equation relating them.
4.3 A proton and an antiproton annihilate. What happens to their combined rest energy?
4.4 Why does the classical formula $E_k = \frac{1}{2}mv^2$ underestimate the kinetic energy of a particle moving at $v = 0.8c$?
5. Match the formula to its meaning
Draw a line (or write the letter) connecting each equation on the left to its correct description on the right. 6 marks (1 each)
| Equation | Description |
|---|---|
| A. $E = \gamma mc^2$ | 1. Rest energy — energy stored in mass at rest. |
| B. $E_k = (\gamma - 1)mc^2$ | 2. Relativistic momentum of a moving particle. |
| C. $E_0 = mc^2$ | 3. Total relativistic energy of a particle. |
| D. $p = \gamma mv$ | 4. Energy due to motion only; zero when at rest. |
| E. $E^2 = (pc)^2 + (mc^2)^2$ | 5. Applies to both massive and massless particles; links $E$, $p$, and $m$. |
| F. $E = pc$ (when $m = 0$) | 6. Relativistic energy of a photon with no rest mass. |
Answers (A–F): A → ___ B → ___ C → ___ D → ___ E → ___ F → ___
6. Label the energy diagram
The diagram below shows how kinetic energy varies with speed for a massive particle. Label the four features A–D using terms from this list: classical $E_k = \frac{1}{2}mv^2$, relativistic $E_k = (\gamma-1)mc^2$, $v = c$ (speed limit), rest energy $mc^2$. 4 marks (1 each)
| Label | Feature name |
|---|---|
| A | |
| B | |
| C | |
| D |
Q1 — Term–definition match
1.1 rest energy • 1.2 Lorentz factor • 1.3 mass-energy equivalence • 1.4 total relativistic energy • 1.5 relativistic kinetic energy • 1.6 relativistic momentum • 1.7 mass defect • 1.8 particle-antiparticle annihilation.
Q2 — True / false with correction
2.1 False. Classical kinetic energy $E_k = \frac{1}{2}mv^2$ is only accurate at speeds much less than $c$. At relativistic speeds the correct formula is $E_k = (\gamma - 1)mc^2$; the classical formula underestimates KE by more than 5% when $\gamma > 1.1$ (i.e. $v \gtrsim 0.4c$).
2.2 False. A particle at rest has non-zero total energy equal to its rest energy $E_0 = mc^2$. Only its kinetic energy is zero when $v = 0$.
2.3 True. As $v \to c$, $\gamma \to \infty$, so $E_k = (\gamma - 1)mc^2 \to \infty$. This is why infinite energy would be required to accelerate a massive particle to $c$.
2.4 False. Massless particles like photons always travel at exactly $c$—not less. Because $m = 0$, the energy-momentum relation $E = pc$ applies; they require no energy input to "push" against a mass.
2.5 True. By definition $E_k = E - E_0 = \gamma mc^2 - mc^2 = (\gamma - 1)mc^2$.
2.6 False. The mass defect is not lost; it is converted to kinetic energy of the reaction products (or photons) in accordance with $E = mc^2$. Conservation of energy is maintained when both mass-energy and kinetic energy are included.
Q3 — Cloze paragraph
In order: Lorentz factor / kinetic / infinite / rest energy / mass-energy equivalence / momentum / massless / gamma-ray photons.
Q4.1 — Why massive particles cannot reach $c$
As a particle's speed approaches $c$, the Lorentz factor $\gamma \to \infty$, so the kinetic energy $E_k = (\gamma - 1)mc^2$ also approaches infinity. An infinite amount of energy would need to be supplied, which is physically impossible, so a massive particle can approach but never reach $c$.
Q4.2 — Total energy vs kinetic energy
Total relativistic energy $E = \gamma mc^2$ includes both rest energy and kinetic energy. Kinetic energy $E_k = (\gamma - 1)mc^2$ is only the energy due to motion. They are related by $E = E_k + mc^2$, i.e. total energy = kinetic energy + rest energy.
Q4.3 — Proton–antiproton annihilation
The combined rest energy (and any kinetic energy) of the proton-antiproton pair is entirely converted into two gamma-ray photons. Each photon carries energy equal to at least $mc^2 = 938$ MeV (the proton rest energy), with any excess kinetic energy also shared between the photons.
Q4.4 — Why classical formula underestimates at $v = 0.8c$
At $v = 0.8c$, $\gamma = 1/\sqrt{1 - 0.64} = 1/0.6 = 1.667$. The relativistic KE is $(\gamma - 1)mc^2 = 0.667mc^2$. The classical KE is $\frac{1}{2}m(0.8c)^2 = 0.32mc^2$. The classical formula assumes mass is constant and momentum grows linearly with speed; it ignores the relativistic increase in inertia as $v \to c$, so it underestimates by about 52%.
Q5 — Formula matching
A → 3 • B → 4 • C → 1 • D → 2 • E → 5 • F → 6.
Q6 — Label the energy diagram
A: Relativistic $E_k = (\gamma - 1)mc^2$ (the steeper blue curve that rises to infinity). B: Classical $E_k = \frac{1}{2}mv^2$ (the dashed grey curve that levels off). C: $v = c$ (speed limit — red dashed vertical line). D: Rest energy $mc^2$ (the $y$-intercept value at zero speed).