Vectors, momentum, energy, orbits, fields and light

Grade 12 Physics

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Forces in Two Dimensions

L1
Lesson 1
Breaking the vector
resolve a force · cos along, sin across
Grade 11 kept forces on one line: a push forward, a weight down. This year they point wherever they like, and the first skill of the course is taking one slanted arrow apart.
The formulas in this lesson
$F_x = F\cos\theta$ , $F_y = F\sin\theta$
Start
L2
Lesson 2
Building the resultant
components in, one net force out
Resolving runs both ways.
The formulas in this lesson
$|\vec{F}| = \sqrt{F_x^2 + F_y^2}$ , $\theta = \tan^{-1}\!\left(\dfrac{F_y}{F_x}\right)$ , $R = \sqrt{A^2 + B^2 + 2AB\cos\theta}$ , $F = ma$
Start
L3
Lesson 3
Friction on the flat
The normal force from the weight
The subscript names the state: k for kinetic, already sliding; s for static, still at rest.
The formula in this lesson
$f_k = \mu_k N$
Start
L4
Lesson 4
Static or kinetic?
the breakaway ceiling and the angle of repose
Lean on a filing cabinet with 20 N and static friction pushes back with exactly 20 N; lean with 60 N and it pushes back with 60 N.
The formulas in this lesson
$f_{s,\max} = \mu_s N$ , $\mu_s = \tan\theta$ , $f_k = \mu_k N$
Start
L5
Lesson 5
The tilted world
The incline's two shadows
Which one takes the cosine is the single most expensive confusion in Grade 12 mechanics, and you never have to memorise it.
The formulas in this lesson
$F_{\parallel} = m g \sin\theta$ , $N = mg\cos\theta$
Start
L6
Lesson 6
Sliding down
a = g sin θ, and what friction takes off it
A feather and an anvil slide down the same frictionless ramp side by side. Galileo's result, arrived at from the algebra rather than the tower.
The formulas in this lesson
$a = g\sin\theta$ , $v = v_0 + at$ , $a = g\left(\sin\theta - \mu_k\cos\theta\right)$
Start
L7
Lesson 7
Around the bend
flat curves, banked curves, and the v² demand
A car at a steady 20 m/s around a bend is accelerating, and this is not a trick of wording. Velocity carries a direction; change the direction and you have changed the velocity, whatever the speedometer says.
The formulas in this lesson
$a_c = \dfrac{v^2}{r}$ , $F_c = \dfrac{m v^2}{r}$ , $v_{\max} = \sqrt{\mu_s g r}$ , $\theta = \tan^{-1}\!\left(\dfrac{v^2}{rg}\right)$
Start
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Boss lesson
The Loading Ramp
the boss · no calculator · weight to normal to friction to speed
Every value is chosen to fit in your head.
The formulas in this lesson
$W = mg$ , $\mu_s = \tan\theta$
Start

Momentum & Collisions

L9
Lesson 9
Momentum itself
p = mv, asked every way
Grade 11 asked what a force does while it acts. Grade 12 asks what it leaves behind, and the answer is momentum: p = mv, read aloud p equals m v.
The formula in this lesson
$p = mv$
Start
L10
Lesson 10
The impulse
A force held for a time
A force on its own changes nothing until it lasts.
The formula in this lesson
$J = F \, \Delta t$
Start
L11
Lesson 11
Impulse meets momentum
FΔt = Δp · the bridge between the two
One warning that costs more marks than any concept here: contact times arrive in milliseconds, and milli means a thousandth — 20 ms = 0.020 s.
Start
L12
Lesson 12
Nothing is lost
conservation of momentum for two bodies
Let two bodies push on each other and nothing else does. Newton's third law makes their forces equal and opposite, over the same contact time, so the impulses are equal and opposite too: whatever one body loses, the other gains.
The formula in this lesson
$m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2$
Start
L13
Lesson 13
Sticking together
the perfectly inelastic collision · one lump, one speed
When two bodies collide and stay together — couplers latching, a bullet in a block, two cars welded by their own bumpers — the collision is perfectly inelastic, and the six-symbol ledger collapses.
The formula in this lesson
$v = \dfrac{m_1 u_1 + m_2 u_2}{m_1 + m_2}$
Start
L14
Lesson 14
The bounce
elastic collisions and the coefficient of restitution
At the other end of the scale sits the perfectly elastic collision, where kinetic energy survives intact along with momentum.
The formulas in this lesson
$v_1 = \dfrac{(m_1 - m_2) u_1 + 2 m_2 u_2}{m_1 + m_2}$ , $e = \dfrac{v_2 - v_1}{u_1 - u_2}$ , $h_2 = e^{2} h_1$
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Boss lesson
The Crash Investigation
the boss · no calculator · momenta before, one lump after
No calculator — the masses are clean tonnes, the speeds whole metres per second, and every line is built to be done in your head.
The formulas in this lesson
$p = mv$ , $a = \dfrac{\Delta v}{\Delta t}$
Start

Energy, All the Way Down

L16
Lesson 16
Work at an angle
Only the along-the-path part counts
Grade 11 asked what forces DO to motion. Grade 12 asks what they COST.
The formula in this lesson
$W = F d \cos\theta$
Start
L17
Lesson 17
Energy of motion
The square does the damage
Solve it for whichever letter the question withholds. Two features, both non-negotiable. The ½ is exact — it falls out of adding up ma over the distance, and a force that grows from zero does, on average, half its maximum.
The formula in this lesson
$E_k = \tfrac{1}{2} m v^{2}$
Start
L18
Lesson 18
Work becomes speed
the work–energy theorem · net work is the KE ledger
Lesson one measured what a force costs. Lesson two measured what motion is worth.
The formula in this lesson
$W_{\text{net}} = \Delta E_k$
Start
L19
Lesson 19
Energy of height
Position as a bank account
Lift something and you have banked energy in it.
The formulas in this lesson
$U = m g h$ , $F_g = m g$
Start
L20
Lesson 20
The spring
The half and the square
Squeeze a spring and it answers in proportion: F = kx — F equals k x. F is the spring force in newtons, x is the displacement from the spring's rest length in metres (stretch or squeeze, the law does not care), and k is the spring constant…
The formulas in this lesson
$U = \tfrac{1}{2} k x^{2}$ , $F = kx$
Start
L21
Lesson 21
The great exchange
conservation · height, speed and spring trade one currency
Height becomes speed; speed becomes squeeze; squeeze becomes height again.
Start
L22
Lesson 22
The rate of doing work
A speedometer for energy
Everything so far totalled energy.
The formulas in this lesson
$P = F v$ , $P = \dfrac{W}{t}$
Start
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Boss lesson
The Coaster Car
the boss · no calculator · hilltop to valley to spring stop
One car, one long ride, and every answer feeds the next: the energy banked at the top, the speed it becomes in the valley, the squeeze that swallows it, the force at full compression — and then the only thing the operator actually cares…
The formula in this lesson
$U = m g h$
Start

Gravitation & Orbits

L24
Lesson 24
The inverse square
Double r, quarter F
Newton's claim was outrageous and correct: the apple and the Moon obey the same rule.
The formula in this lesson
$F = \dfrac{G m_1 m_2}{r^2}$
Start
L25
Lesson 25
Little g from big G
Weight is a field reading
Keep the three symbols apart, because an exam will absolutely try to blur them.
The formula in this lesson
$g = \dfrac{GM}{r^2}$
Start
L26
Lesson 26
Falling around the Earth
R runs from the centre
A satellite is not beyond gravity; it is falling continuously and moving sideways fast enough to keep missing.
The formulas in this lesson
$v = \sqrt{\dfrac{GM}{r}}$ , $F_c = \dfrac{m v^{2}}{r}$
Start
L27
Lesson 27
The year of a satellite
Two roads
The period T is the time for one full circuit, in seconds. There are two honest roads to it, and the givens decide which one you walk.
The formulas in this lesson
$v = \dfrac{2\pi r}{T}$ , $T = 2\pi\sqrt{\dfrac{r^3}{GM}}$
Start
L28
Lesson 28
Kepler's bargain
T² over r³ · one ratio rules every moon
Subscript 1 is the body you are solving for; subscript 2 is the one you already know.
The formula in this lesson
$\dfrac{T_1^2}{T_2^2} = \dfrac{a_1^3}{a_2^3}$
Start
L29
Lesson 29
Leaving for good
escape velocity · and the negative well
Gravitational potential energy near a surface is U = mgh — U equals m g h, with m the mass in kilograms, g the local field strength, and h the height above a zero you chose yourself (the bench, the floor, the ground).
The formulas in this lesson
$v = \sqrt{\dfrac{2GM}{r}}$ , $U = -\dfrac{GMm}{r}$
Start
★
Boss lesson
Mission Control
the boss · no calculator · field, speed, period, escape
Split the mantissa from the exponent, take the root of each, put them back together. Powers of ten are furniture here, not obstacles.
The formula in this lesson
$g = \dfrac{GM}{r^2}$
Start

Electric & Magnetic Fields

L31
Lesson 31
Coulomb's count
the electric inverse square · microcoulombs and the constant
You have met this shape before.
The formulas in this lesson
$F = \dfrac{k_e q_1 q_2}{r^2}$ , $Q = It$
Start
L32
Lesson 32
The charge reservoir
C = Q/V and the energy a capacitor banks
A capacitor is two plates that hold charge apart.
The formulas in this lesson
$C = \dfrac{Q}{V}$ , $E = \tfrac{1}{2} C V^{2}$
Start
L33
Lesson 33
The sideways push
The force that only notices motion
Electric fields push any charge, moving or not. Magnetic fields are pickier: they only notice charge that is MOVING, and they push it sideways.
The formula in this lesson
$F = q v B \sin\theta$
Start
L34
Lesson 34
Wires in the field
The motor effect · and wires pushing each other
Direction, in one line: currents running the SAME way attract, opposite ways repel — the opposite of what charges do, and worth saying out loud twice.
The formulas in this lesson
$F = B I L \sin\theta$ , $F = \dfrac{\mu_0 I_1 I_2 \ell}{2\pi d}$
Start
L35
Lesson 35
The flux through the loop
solenoid fields · Φ = BA cos θ · and the weber
Look at what that means: N/L is turns per metre, so the field only cares how tightly the turns are packed, never how many there are in total.
The formulas in this lesson
$B = \dfrac{\mu_0 N I}{L}$ , $\Phi = B A \cos\theta$
Start
L36
Lesson 36
The changing flux
Faraday's law and motional EMF · change is what induces
Read the formula as the sentence it is: a coil generates voltage only while the flux through it is CHANGING. Park a coil in the strongest steady field on Earth and it produces exactly zero.
The formulas in this lesson
$\varepsilon = N \dfrac{\Delta\Phi}{\Delta t}$ , $\varepsilon = B L v$ , $\varepsilon = 0$
Start
★
Boss lesson
The Mass Spectrometer
the boss · no calculator · qvB supplies mv²/r
Here is the composition the whole chapter has been walking toward. The magnetic force F = qvB is the only sideways actor in the chamber, and it is always perpendicular to the motion — so it never changes the ion's speed, only its direction.
The formula in this lesson
$F = q v B \sin\theta$
Start

The Wave Nature of Light

L38
Lesson 38
The wave equation
Nanometres, terahertz, and the speed limit
Read aloud: v equals f lambda.
The formulas in this lesson
$T = \dfrac{1}{f}$ , $v = f\lambda$
Start
L39
Lesson 39
Slower in glass
Light's speed tax, and what survives the crossing
Nothing outruns light in a vacuum.
The formula in this lesson
$n = \dfrac{c}{v}$
Start
L40
Lesson 40
The bending rule
Snell's law · sines, not angles · measured from the normal
Light crossing a boundary changes speed, and a wave that changes speed at a slant changes direction.
The formula in this lesson
$n_1 \sin\theta_1 = n_2 \sin\theta_2$
Start
L41
Lesson 41
Trapped light
the critical angle and apparent depth · refraction's two tricks
Push Snell's law and it breaks in an interesting way.
The formulas in this lesson
$\sin\theta_c = \dfrac{n_2}{n_1}$ , $d' = \dfrac{d}{n}$
Start
L42
Lesson 42
Two slits, one pattern
Measuring light with a ruler · the unit gauntlet
In 1801 Thomas Young let one beam of light through two narrow slits and caught the result on a screen. Not two bright lines — a whole row of them, evenly spaced, with darkness in between.
The formula in this lesson
$\Delta y = \dfrac{\lambda L}{d}$
Start
L43
Lesson 43
The grating
Orders, line counts, and the ceiling on sin θ
Two slits give soft fringes. Thousands of them, ruled side by side on one piece of glass, give razor-thin beams at precise angles — a diffraction grating, the instrument that reads the chemistry of a star from its light.
The formula in this lesson
$m\lambda = d\sin\theta$
Start
L44
Lesson 44
The polarizing angle
Brewster's angle · the one formula in the chapter with a tangent
Light reflecting off a flat surface comes back partly polarized — its vibrations biased into one plane.
The formula in this lesson
$\tan\theta_B = \dfrac{n_2}{n_1}$
Start
L45
Lesson 45
Colours in the film
thin-film interference · why soap bubbles wear rainbows
A soap bubble is colourless stuff. The rainbow on it comes from arithmetic.
The formulas in this lesson
$2nt = \left(m + \tfrac{1}{2}\right)\lambda$ , $2nt = m\lambda$
Start
★
Boss lesson
The Optics Bench
the boss · no calculator · one laser, four instruments
One laser travels the whole length of the course: the wave equation gives its frequency, the index gives its speed and wavelength inside a block, Snell's law bends it at the face, and Young's slits paint what is left on the wall.
The formula in this lesson
$v = f\lambda$
Start

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