# Introduction to Modern Physics

Mathematics I · Physics · https://tryals.app/learn/mathematics-i/introduction-to-modern-physics

## Beyond the Everyday

By 1900, classical physics seemed complete until **relativity** and **quantum mechanics** overturned it at extremes. **Special relativity** (1905) rests on one postulate: the **speed of light is the same for all observers**. Space and time must adjust to keep this true.

| Phenomenon | Definition |
| --- | --- |
| **Time dilation** | A moving clock runs slow relative to a stationary observer. |
| **Length contraction** | A moving object is shortened along its direction of motion. |
| **Mass-energy equivalence** | **$E = mc^2$** where $E$ is energy, $m$ is mass, and $c$ is the speed of light. |

**General relativity** extends this by reinterpreting gravity as the **curvature of spacetime** by mass. These effects dominate near light speed but are negligible at everyday speeds.

## The Quantum Realm

**Quantum mechanics** governs the atomic scale with stranger rules. Energy comes in discrete packets called **quanta**, and light and matter exhibit **wave-particle duality**.

| Quantum Principle | Rule & Implication |
| --- | --- |
| **Uncertainty principle** | You cannot know a particle's position and momentum both precisely. |
| **Probability** | The theory predicts statistical outcomes, not certainties, using a wavefunction. |

> **Common pitfall:** The **uncertainty principle** is a fundamental limit of nature, not a flaw from clumsy measurement. **Quantum mechanics** is inherently **probabilistic**, a departure from deterministic classical physics.

## Practice questions

7 of this lesson's 13 practice questions, with answers. The full set is in the app.

### 1. Order the logical chain of special relativity.

**Answer:**

1. Take it as given that every observer measures light at the same speed
2. Notice this is impossible if time and length are the same for everyone
3. Accept that measurements of time and length must depend on the observer's motion
4. Derive time dilation and length contraction as the consequences
5. Deduce that mass is itself a form of energy, E = mc squared

**Why:** Everything follows from one postulate plus a refusal to tolerate contradiction. The famous mass-energy relation is a late consequence, not a starting point, which is why relativity is a good example of physics being deductive rather than a collection of separate discoveries.

Page: https://tryals.app/practice/mathematics-i/introduction-to-modern-physics/order-the-logical-chain-of-special-relativity

### 2. A transformer changes voltage using no moving parts and cannot work on direct current. Why does DC defeat it?

A. Because the iron core saturates immediately under DC
B. Because transformers require the two coils to be electrically connected, which DC prevents
C. Because induction responds to a changing flux, and DC is steady
D. Because DC cannot flow through a coil of wire

**Answer:** C. Because induction responds to a changing flux, and DC is steady

**Why:** Faraday's law pays for change, and DC offers none, steady flux induces nothing. That single fact settled the war of the currents: AC could be transformed to high voltage for efficient transmission and back down for use, and DC could not, which is why the grid is AC.

Page: https://tryals.app/practice/mathematics-i/introduction-to-modern-physics/a-transformer-changes-voltage-using-no-moving-parts-and-cannot-work

### 3. Special relativity says the speed of light is the same for every observer, whatever their own motion. Why does that single postulate force time and length to be observer-dependent?

A. Because measuring instruments become less accurate at high speeds.
B. Because the postulate is an approximation, and the discrepancies are what appear as time dilation.
C. Because light exerts a force on clocks and rulers, physically altering them.
D. Because speed is distance over time, so those must be what varies.

**Answer:** D. Because speed is distance over time, so those must be what varies.

**Why:** The argument is almost arithmetic: speed is distance over time, and if two observers in relative motion both measure c, their distances and times cannot both agree. Nothing is happening *to* the clocks, a moving clock is in perfect order in its own frame. And the effects are routine: atmospheric muons reach the ground only because their lifetimes are dilated.

Page: https://tryals.app/practice/mathematics-i/introduction-to-modern-physics/special-relativity-says-the-speed-of-light-is-the-same-for-every

### 4. According to quantum mechanics, a particle always has a definite position and a definite momentum, and the uncertainty principle merely says we cannot measure both at once.

**Answer:** False

**Why:** False. The uncertainty principle is not a statement about measurement clumsiness or hidden knowledge: position and momentum cannot both have precise values simultaneously. Treating it as ignorance of pre-existing values is the most common misreading of the whole theory.

Page: https://tryals.app/practice/mathematics-i/introduction-to-modern-physics/according-to-quantum-mechanics-a-particle-always-has-a-definite

### 5. Take the speed of light as 3 x 10^8 m/s. Fully converting 2 kg of matter to energy gives E = 1.8 x 10^n joules. Enter the value of n.

**Answer:** 17

**Why:** $c^{2} = 9 \times 10^{16}$, so $E = 2 \times 9 \times 10^{16} = 18 \times 10^{16} = 1.8 \times 10^{17}$ J. The renormalisation from 18 to 1.8 bumps the exponent by one, a step routinely dropped, giving 16. The enormous size of the number is why a tiny mass powers a reactor.

Page: https://tryals.app/practice/mathematics-i/introduction-to-modern-physics/take-the-speed-of-light-as-3-x-10-8-m-s-fully-converting-2-kg-of

### 6. Select every statement that correctly describes the uncertainty principle.

A. It concerns position and momentum held simultaneously
B. It is a fundamental limit of nature, not a limitation of equipment
C. It could be overcome with sufficiently good instruments
D. It arises because measurement instruments are clumsy and disturb the particle
E. The more precisely position is known, the less precisely momentum can be

**Answer:** A. It concerns position and momentum held simultaneously; B. It is a fundamental limit of nature, not a limitation of equipment; E. The more precisely position is known, the less precisely momentum can be

**Why:** The principle says position and momentum cannot both have precise values at once, it is a statement about nature, not about laboratory technique. Options 1 and 4 are the 'clumsy measurement' misreading; no improvement in instruments could ever evade the limit.

Page: https://tryals.app/practice/mathematics-i/introduction-to-modern-physics/select-every-statement-that-correctly-describes-the-uncertainty

### 7. Complete the founding postulate of special relativity.

**Answer:** Special relativity begins from the postulate that **the speed of light** is the same for every observer, whatever their state of motion.

**Why:** Holding light speed absolute is what forces time and length to become relative, the three distractors are the very quantities that had to give way. This inversion of what counts as absolute is the whole content of the theory.

Page: https://tryals.app/practice/mathematics-i/introduction-to-modern-physics/complete-the-founding-postulate-of-special-relativity