Topic 1

Classical Mechanics

The study of motion and the forces that change it. From falling apples to orbiting planets, mechanics is the oldest and most intuitive branch of physics — yet it underpins every other topic in this guide.

Kinematics — describing motion
Newton's laws of motion
Energy & work
Momentum & impulse
Rotational motion
Oscillations & waves

1. Kinematics — Describing Motion

Kinematics is the language of motion. We describe where something is using position $\vec{r}(t)$, how fast it's moving with velocity $\vec{v} = d\vec{r}/dt$, and how the velocity itself changes with acceleration $\vec{a} = d\vec{v}/dt$.

$$\vec{v}(t) = \vec{v}_0 + \int_0^t \vec{a}\,dt',\qquad \vec{r}(t) = \vec{r}_0 + \int_0^t \vec{v}(t')\,dt' $$

Constant acceleration — the most common case, like free fall. With $a$ constant, the four kinematic equations follow by integration. They only apply when acceleration never changes direction or magnitude.

For motion in a straight line with constant acceleration $a$, starting from position $x_0$ with initial velocity $v_0$:

$$x(t) = x_0 + v_0 t + \tfrac{1}{2} a t^2$$

$$v(t) = v_0 + a t$$

The last Suvat equation connects the others without $t$:

$$v^2 = v_0^2 + 2 a (x - x_0)$$

Example. A ball is dropped from a height of 45 m. How long until it hits the ground? Take $g = 9.8\,\mathrm{m/s^2}$ and $v_0 = 0$.
Using $x = x_0 + v_0 t + \frac{1}{2}at^2$ with $x = 0$: $\;0 = 45 - \frac{1}{2}(9.8)t^2$, giving $t = \sqrt{2 \times 45 / 9.8} \approx 3.03$ s.

Projectile motion

When we launch an object, the only acceleration is gravity downward, so horizontal velocity stays constant while vertical velocity changes uniformly:

$$x(t) = v_0\cos\theta\; t,\qquad y(t) = v_0\sin\theta\; t - \tfrac{1}{2}g t^2$$

Drag the sliders below the simulation to launch a projectile at different speeds and angles. Can you find the angle that maximizes range?

Maximum range occurs at $\theta = 45^\circ$ (when launch and landing heights match) because $\sin 2\theta$ peaks at $90^\circ$.

2. Newton's Laws of Motion

In 1687, Newton laid down three rules that still govern engineering today:

Newton's First Law (Inertia) — A body at rest stays at rest, and a body in motion stays in uniform straight-line motion, unless a net external force acts on it.

Newton's Second Law — The rate of change of momentum equals the net force: $\vec{F}_{\text{net}} = \dfrac{d\vec{p}}{dt}$. For constant mass this becomes $\vec{F} = m\vec{a}$.

Newton's Third Law — For every action there is an equal and opposite reaction: $\vec{F}_{AB} = -\vec{F}_{BA}$.

$$\boxed{\vec{F}_{\text{net}} = m\,\vec{a}}$$

Concept	Symbol	Units	Typical example
Force	$F$	N (kg·m/s²)	Weight of 1 kg mass
Mass	$m$	kg	Inertia to acceleration
Acceleration	$a$	m/s²	Free-fall: 9.8 m/s²
Momentum	$p$	kg·m/s	Moving car

Caution: Newton's laws are valid only in inertial reference frames. In an accelerating car, a "mysterious" force flings you backward — but in an inertial frame that's just your inertia.

3. Energy & Work

Work is force times the distance over which it acts: $W = \vec{F}\cdot\vec{d} = Fd\cos\theta$. It's the bridge between force and energy.

$$W = \int \vec{F}\cdot d\vec{r}$$

Two forms of mechanical energy dominate classical mechanics:

Kinetic energy: $K = \tfrac{1}{2}mv^2$ — energy of motion.
Gravitational potential energy: $U = mgh$ (near Earth's surface) or $U = -GMm/r$ (general).

Conservation of Energy — In an isolated system, total energy $E = K + U$ is constant. Energy cannot be created or destroyed; it only transforms between forms.

$$K_1 + U_1 = K_2 + U_2$$

This is powerful because it sidesteps tricky force vectors — we just track energy before and after.

4. Momentum & Impulse

Momentum $p = mv$ captures "how hard it is to stop" a moving object. Impulse $J = \Delta p = \vec{F}_{\text{avg}}\,\Delta t$ is the push that changes momentum.

$$\vec{J} = \Delta \vec{p} = \int \vec{F}\,dt$$

Conservation of Momentum — If $\vec{F}_{\text{net, ext}} = 0$, total momentum of a system is conserved: $\sum \vec{p}_{\text{initial}} = \sum \vec{p}_{\text{final}}$.

$$m_1 \vec{v}_{1i} + m_2 \vec{v}_{2i} = m_1 \vec{v}_{1f} + m_2 \vec{v}_{2f}$$

This holds even in collisions where kinetic energy is lost to heat or deformation — as long as no external force acts.

5. Rotational Motion

Rotational motion mirrors linear motion with angular analogs. Angle $\theta$ replaces position, angular velocity $\omega = d\theta/dt$ replaces $v$, and angular acceleration $\alpha = d\omega/dt$ replaces $a$.

$$v = r\omega,\qquad a_t = r\alpha,\qquad a_c = r\omega^2$$

The rotational analog of $F = ma$ uses moment of inertia $I$ and torque $\tau$:

$$\tau = I\alpha$$

Torque = force applied at a distance from the pivot. A longer wrench produces more torque for the same force.

Common moments of inertia:

Object	Axis	$I$
Solid sphere	Through center	$\frac{2}{5}MR^2$
Hollow sphere	Through center	$\frac{2}{3}MR^2$
Solid cylinder	Through center	$\frac{1}{2}MR^2$
Thin rod	Through center, ⊥ rod	$\frac{1}{12}ML^2$
Thin rod	Through end, ⊥ rod	$\frac{1}{3}ML^2$

6. Oscillations & Simple Harmonic Motion

Simple harmonic motion (SHM) describes systems where a restoring force is proportional to displacement: $F = -kx$. The solution is sinusoidal:

$$x(t) = A\cos(\omega t + \phi),\qquad \omega = \sqrt{\frac{k}{m}}$$

$$T = \frac{2\pi}{\omega} = 2\pi\sqrt{\frac{m}{k}},\qquad f = \frac{1}{T}$$

where $T$ is the period, $f$ the frequency, $A$ the amplitude, and $\phi$ a phase offset.

A simple pendulum. Adjust length, gravity, and damping to see how the period $T = 2\pi\sqrt{L/g}$ changes.

Small-angle approximation. A real pendulum only obeys $T = 2\pi\sqrt{L/g}$ when the swing angle is small (sin θ ≈ θ). Large swings are slower than the formula predicts — the simulation captures this.

Waves

Many oscillators couple into waves. A traveling wave has the form $y(x,t) = A\sin(kx - \omega t + \phi)$, where $k = 2\pi/\lambda$ is the wave number and $\omega = 2\pi f$. The wave speed is $v = f\lambda = \omega/k$.

$$y(x,t) = A\sin\!\left(\frac{2\pi}{\lambda}x - 2\pi f\, t + \phi\right)$$

In a standing wave, reflected waves interfere with incoming ones. Nodes (zero displacement) appear at $x = n\lambda/2$. This is why guitar strings of fixed length only play certain notes — the boundary condition selects allowed wavelengths.

Key Takeaways

Motion is described with position, velocity, and acceleration — the derivatives of $\vec{r}(t)$.
$\vec{F} = m\vec{a}$ is the master equation of mechanics, but only in inertial frames.
Energy and momentum are conserved in isolated systems — powerful shortcuts past force diagrams.
Rotational motion reuses linear formulas with $I$, $\tau$, and $\omega$ replacing $m$, $F$, $v$.
Simple harmonic oscillators ($F = -kx$) and waves ($v = f\lambda$) are everywhere in nature.

🔥 Next: Thermodynamics →

Live seismology

Real earthquakes recorded in the last 24 hours, magnitude 4.5 and above.

🌍 Recent significant earthquakes

Source: USGS Earthquake Hazards Program — free, no key required.

Contacting the USGS earthquake feed…