What you’ll practice

• SI units, vectors, significant figures, uncertainties
• 1D motion & projectile motion, graphs
• Forces, Newton’s laws, friction, collisions

How to use

• Use sliders/inputs then press ▶ to animate.
• Hover tooltips for notes. Use R to reset many widgets.
• Print to PDF with the top‑right button.

Assumptions

• Flat Earth (local lab scale), constant g.
• No air resistance unless stated.
• SI units throughout (unless noted).

Vectors

Magnitude: \( |\vec r| = \sqrt{x^{2}+y^{2}} \)   ·   Direction: \( \theta = \operatorname{atan2}(y,\,x) \)

Addition: \( \vec r = \vec a + \vec b \)

Uncertainties

Add/Subtract: absolute add → \( \Delta z \approx \Delta x + \Delta y \)

× / ÷: % add → \( \dfrac{\Delta z}{z} \approx \dfrac{\Delta x}{x} + \dfrac{\Delta y}{y} \)

Powers: \( \dfrac{\Delta (x^{n})}{x^{n}} \approx |n|\dfrac{\Delta x}{x} \)

Round \( \Delta \) to 1–2 s.f.; match value to \( \Delta \)’s decimal places.

Dynamics

Newton II: \( \sum F = m a \)   ·   Momentum: \( p = m\,v \)   ·   Work: \( W = F\,s \)

Static friction: \( F_f \le \mu_s N \)   ·   Kinetic: \( F_f = \mu_k N \)

Energy & Power

Kinetic: \( E_k = \tfrac{1}{2} m v^{2} \)

Gravitational: \( E_p = m g h \)

Work: \( W = F s \cos\theta \)

Power: \( P = \dfrac{W}{t} = F\,v \)

Efficiency: \( \eta = \dfrac{P_{out}}{P_{in}} \)

Momentum & Impulse

Momentum: \( p = m\,v \)

Impulse: \( J = F\,\Delta t = \Delta p \)

Conservation: \( \sum p_{\text{before}} = \sum p_{\text{after}} \)

Restitution: \( e = \dfrac{v'_2 - v'_1}{v_1 - v_2} \)

Circular Motion

Centripetal: \( a_c = \dfrac{v^{2}}{r} = \omega^{2} r \)

Force: \( F_c = m \dfrac{v^{2}}{r} \)

Link: \( v = \omega r,\; T = \dfrac{2\pi}{\omega} = \dfrac{2\pi r}{v} \)

Simple Harmonic Motion

Definition: \( a = -\omega^{2} x \)

Displacement: \( x = A\cos(\omega t + \phi) \)

Speed: \( v = \pm\,\omega\sqrt{A^{2}-x^{2}} \)

Spring: \( T = 2\pi\sqrt{\tfrac{m}{k}} \)

Pendulum: \( T = 2\pi\sqrt{\tfrac{\ell}{g}} \)

Waves

Wave speed: \( v = f\lambda \)

Intensity: \( I \propto A^{2} \)

Young: \( d\sin\theta = m\lambda \) maxima

Grating: \( n\lambda = d\sin\theta \)

Phase: \( \phi = \dfrac{2\pi \Delta s}{\lambda} \)

Optics

Snell: \( n_1\sin\theta_1 = n_2\sin\theta_2 \)

Critical: \( \sin\theta_c = \dfrac{n_2}{n_1} \) (\(n_1>n_2\))

Thin lens: \( \dfrac{1}{f} = \dfrac{1}{u} + \dfrac{1}{v} \)

Magnification: \( m = \dfrac{v}{u} = \dfrac{h'}{h} \)

Electricity

Ohm: \( V = I R \)

Power: \( P = VI = I^{2}R = \dfrac{V^{2}}{R} \)

Resistivity: \( R = \rho \dfrac{L}{A} \)

Series: \( R_{eq} = R_1+R_2+\dots \)

Parallel: \( \dfrac{1}{R_{eq}} = \sum \dfrac{1}{R_i} \)

EMF: \( \varepsilon = V + I r \)

Capacitors

Definition: \( C = \dfrac{Q}{V} \)

Series: \( \dfrac{1}{C_{eq}} = \sum \dfrac{1}{C_i} \)

Parallel: \( C_{eq} = C_1+C_2+\dots \)

Energy: \( U = \tfrac{1}{2} C V^{2} = \tfrac{1}{2} QV = \dfrac{Q^{2}}{2C} \)

RC: \( V(t)=V_0\!(1-e^{-t/RC}),\; V_{dis}(t)=V_0 e^{-t/RC} \)

Fields & Induction

Gravity: \( g=\dfrac{GM}{r^{2}},\; V_g=-\dfrac{GM}{r} \)

Electric: \( E=\dfrac{kQ}{r^{2}},\; V_e=\dfrac{kQ}{r},\; F=q\,E \)

Magnetic: \( F=B I L\,\sin\theta= B q v\,\sin\theta \)

Radius: \( r=\dfrac{m v}{|q| B} \)

Flux/EMF: \( \Phi=BA\cos\theta,\; |\mathcal{E}|=N\,\dfrac{d\Phi}{dt} \)

Thermal

Ideal gas: \( pV=nRT = NkT \)

Internal energy: monoatomic \( U=\tfrac{3}{2}nRT \)

Heating: \( Q=mc\,\Delta T \), Latent: \( Q=mL \)

First law: \( \Delta U = Q - W \)

Quantum

Photon: \( E=hf=\dfrac{hc}{\lambda} \)

Photoelectric: \( hf = \phi + K_{\max} \)

de Broglie: \( \lambda = \dfrac{h}{p} = \dfrac{h}{mv} \)

Nuclear

Decay: \( N = N_0 e^{-\lambda t} \), \( A=\lambda N \)

Half-life: \( T_{1/2} = \dfrac{\ln 2}{\lambda} \)

Mass–energy: \( E = \Delta m\,c^{2} \)

Microscopy

Magnification: \( M = \dfrac{I}{A} \)

Scale bar: \( \text{size} = \dfrac{\text{measured}}{\text{scale}} \)

Resolution limit ≈ \( \tfrac{\lambda}{2} \) (light)

Transport & Exchange

Fick: \( \text{rate} \propto \dfrac{A\,\Delta C}{d} \)

Cardiac output: \( \text{CO} = \text{SV} \times \text{HR} \)

Ventilation rate: \( f = \dfrac{\Delta V}{\Delta t} \)

Water Potential

Total: \( \Psi = \Psi_s + \Psi_p \)

Solute: \( \Psi_s = -i C R T \) (approx.)

Osmosis to lower \(\Psi\)

Population & Diversity

Simpson: \( D = 1 - \sum \left(\dfrac{n}{N}\right)^2 \)

Mark–release–recapture: \( \hat N = \dfrac{n_1 n_2}{m} \)

Enzymes

Rate: \( v = \dfrac{\Delta P}{\Delta t} \)

Michaelis–Menten (qual.): \( v = \dfrac{V_{max}[S]}{K_M + [S]} \)

Q10: rate change per 10°C

Statistics (Core)

Mean: \( \bar x = \dfrac{\sum x}{n} \)

SD: \( s = \sqrt{\dfrac{\sum (x-\bar x)^2}{n-1}} \)

t‑test (two‑sample): \( t = \dfrac{\bar x_1-\bar x_2}{s_p\sqrt{\tfrac{1}{n_1}+\tfrac{1}{n_2}}} \)

Amounts & Solutions

Moles: \( n = \dfrac{m}{M_r} \)

Concentration: \( c = \dfrac{n}{V} \)

Dilution: \( c_1 V_1 = c_2 V_2 \)

Gases & Energy

Ideal gas: \( pV = nRT \)

Heat: \( q = mc\,\Delta T \)

Enthalpy change per mol: \( \Delta H = -\dfrac{q}{n} \)

Stoichiometry

% yield: \( \dfrac{\text{actual}}{\text{theoretical}} \times 100\% \)

Atom economy: \( \dfrac{\text{Mr desired}}{\sum \text{Mr products}} \times 100\% \)

Acids & pH

\( \mathrm{pH} = -\log_{10}[\mathrm{H}^+] \)

\( K_a = \dfrac{[\mathrm{H}^+][\mathrm{A}^-]}{[\mathrm{HA}]} \)

\( K_w = [\mathrm{H}^+][\mathrm{OH}^-] = 1.0\times10^{-14} \)

Equilibria

\( K_c = \dfrac{\prod [\text{products}]^{\nu}}{\prod [\text{reactants}]^{\nu}} \)

Reaction quotient \(Q\) same form as \(K\)

Le Châtelier principle (qual.)

Kinetics

Rate law: \( r = k\,[A]^m[B]^n \)

Arrhenius: \( k = A e^{-E_a/(RT)} \)

Half‑life (1st order): \( t_{1/2} = \dfrac{\ln 2}{k} \)

Electrochemistry

\( E_{cell}^{\circ} = E_{red}^{\circ} - E_{ox}^{\circ} \)

\( \Delta G^{\circ} = -n F E_{cell}^{\circ} \)

Thermodynamics

\( \Delta G = \Delta H - T\,\Delta S \)

\( K \approx e^{-\Delta G^{\circ}/(RT)} \)

Titrations

For monoprotic/1:1: \( c_a V_a = c_b V_b \)

Use stoichiometric ratios when not 1:1