Interactive resources
See the physics — and the mathematics beneath it
Live, adjustable models across the IB syllabus. Change the parameters, watch what changes, and read the examiner's note on where marks are won and lost.
A.1 Projectile motion SL & HL
Fire a projectile and vary the conditions. Switch on air resistance, change gravity, or launch from a height — the horizontal and vertical motions stay independent.
Range
Max height
Time of flight
Examiner's note
The x and y motions are independent. Without air resistance the range peaks at 45°; with it, the optimum angle drops and the path is no longer symmetric. Quote your value of g, show the method, and give a sensible number of significant figures.
C.3 Wave superposition SL & HL
Two travelling waves add point by point. Change each wave's amplitude and wavelength, shift the phase, and watch the resultant.
Resultant
Examiner's note
The principle of superposition: the resultant displacement is the algebraic sum of the individual displacements. Equal wavelengths in phase interfere constructively; a 180° phase difference cancels. Watch for the mark that asks why, not just what.
C.1 Simple harmonic motion SL & HL
A mass on a spring. The period depends on mass and stiffness, not amplitude. Add damping and watch the amplitude decay within its envelope.
Period T
Displacement
Velocity
Examiner's note
The defining relation is a = −ω²x, with T = 2π√(m/k) — independent of amplitude. Velocity is greatest at equilibrium and zero at the extremes. Damping removes energy, shrinking amplitude while the period stays almost unchanged.
D.2 Electric field of point charges SL & HL
Each arrow shows the force on a positive test charge. Drag the charges, change their strength, or switch to a potential map — the field is always the vector sum.
Examiner's note
Field lines run from positive to negative, never cross, and are densest where the field is strongest. Potential is a scalar — it adds arithmetically; field is a vector — it adds by components. Confusing the two is a frequent lost mark.
A.2 Circular motion SL & HL
Constant speed, changing velocity. The centripetal force points inward; increase the mass or speed, or tighten the radius, and read how the force responds.
Period T
Accel. v²/r
Force mv²/r
ω
Examiner's note
Speed is constant but velocity is not — direction changes, so there is acceleration. F = mv²/r points to the centre. A classic lost mark: treating the outward "centrifugal force" as real. It is not.
5 Tangents & the derivative AA & AI · SL & HL
The gradient at a point is the derivative. Drag the point, and switch on the derivative curve f′(x) to see where the gradient is zero, positive or negative.
Point
Gradient f′(x)
Examiner's note
"Find the gradient" rewards the chain of reasoning. Where f′(x) = 0 you have a stationary point; the sign of f′ tells you increasing or decreasing. Showing the derivative curve makes the connection examiners test explicit.
2 Function transformations AA & AI · SL & HL
Build y = a·f(b(x − c)) + d from a base function. Each parameter stretches, shifts or reflects the graph in a distinct way.
Examiner's note
Order matters: b and c act inside the bracket (horizontal, and counter-intuitive — a bigger b compresses), while a and d act outside (vertical). Negative a or b gives a reflection. A frequent lost mark is shifting the wrong way, or in the wrong order.
4 Scatter & regression AA & AI · SL & HL
The least-squares line minimises the vertical distances to the points. Drag any point, add your own by clicking, and watch r respond. Show the residuals to see what "least squares" means.
Regression line
Correlation r
r²
Examiner's note
The regression line of y on x minimises vertical residuals — use it to predict y from x, not the reverse. Quote r for strength and r² for the proportion of variation explained. Remember: correlation is not causation, and extrapolation is risky.
B.3 Ideal gas laws SL & HL
Pressure, volume and temperature are linked by PV = nRT. Change one and watch the others — and the particles — respond.
Pressure P
Volume V
Temperature T
Examiner's note
At constant T, P ∝ 1/V (Boyle). At constant P, V ∝ T in kelvin (Charles). Temperature must be in kelvin — a very common lost mark. Particle speed rises with T; pressure comes from their collisions with the walls.
E.3 Radioactive decay SL & HL
Decay is random for any one nucleus, but predictable in bulk. Each half-life halves what remains — watch the curve and the sample together.
Remaining
Half-lives elapsed
Fraction left
Examiner's note
Decay is exponential: N = N₀e^(−λt), with λ = ln2 / t½. After one half-life, half remains; after two, a quarter — never zero in theory. The randomness is why we speak of probability, not certainty, for a single nucleus.
D.3 Force on a current-carrying wire SL & HL
A wire carrying a current in a magnetic field feels a force — the motor effect. The force is greatest when the wire is perpendicular to the field.
Force F = BIL sinθ
Examiner's note
F = BIL sinθ, where θ is the angle between the current and the field. Force is maximum at 90° and zero when the wire is parallel to the field. Use Fleming's left-hand rule for the direction — a frequent exam step.
5 Area under a curve AA & AI · SL & HL
The definite integral is the exact area. Rectangles estimate it — add more, and the estimate closes in on the true value.
Rectangle estimate
Exact integral
Error
Examiner's note
The definite integral ∫ₐᵇ f(x) dx is the signed area between the curve and the x-axis. Rectangles are a numerical estimate that improves as n grows — the idea behind the trapezoidal rule you meet in IB. Watch the units and the limits.
3 The unit circle AA & AI · SL & HL
Sine and cosine are the vertical and horizontal coordinates of a point on the unit circle. Drag the angle and watch the wave unroll.
sin θ
cos θ
tan θ
Examiner's note
On the unit circle a point is (cos θ, sin θ). This is why sine and cosine repeat every 360° and why sin²θ + cos²θ = 1. Knowing the exact values at 30°, 45° and 60° saves time and marks in non-calculator papers.
4 The normal distribution AA & AI · SL & HL
A bell curve set by its mean and standard deviation. Shade a region to read the probability — the area under the curve.
P(a < X < b)
z (lower)
z (upper)
Examiner's note
Probability is the area under the curve. Standardise with z = (x − μ)/σ to compare or to use tables. Roughly 68% lies within one σ of the mean, 95% within two — a check worth remembering.
C.4 Standing waves on a string SL & HL
Fixed at both ends, a string vibrates only at certain frequencies — the harmonics. Nodes stay still; antinodes swing the most.
Nodes
Antinodes
Wavelength
Examiner's note
A standing wave forms when two identical waves travel in opposite directions and superpose. Only wavelengths λ = 2L/n fit a string fixed at both ends. Nodes have zero amplitude; energy is not transferred along the string — a distinction examiners test.
C.5 The Doppler effect SL & HL
A moving source bunches its wavefronts ahead and stretches them behind — raising the frequency you hear approaching, lowering it receding.
Emitted f
Heard ahead
Examiner's note
For a moving source, f′ = f·v/(v ∓ vₛ) — minus when approaching (higher pitch), plus when receding. The wave speed in the medium is unchanged; it is the wavelength that compresses. Be careful which speed moves: source or observer.
E.2 The photoelectric effect HL
Light ejects electrons only above a threshold frequency. Below it, no amount of intensity helps — evidence that light comes in quanta.
Max KE
Threshold f₀
Emission?
Examiner's note
E = hf and KEₘₐₓ = hf − φ. Below the threshold frequency f₀ = φ/h, no electrons escape, whatever the intensity. Intensity changes the number of electrons, not their energy — the observation classical physics could not explain.
4 The binomial distribution AA & AI · SL & HL
Count successes in n independent trials, each with probability p. The bars give P(X = k) — highlight one to read its probability.
P(X = k)
Mean np
Examiner's note
P(X = k) = ⁿCₖ pᵏ(1−p)ⁿ⁻ᵏ, with mean np and variance np(1−p). The conditions matter: fixed n, independent trials, constant p, two outcomes. State them — it is often a mark in itself.
1 Complex numbers · the Argand plane AA HL & AI HL
A complex number is a point: real part across, imaginary part up. Drag z and read its modulus and argument — or see z² rotate and scale.
z
Modulus |z|
Argument
Examiner's note
In modulus–argument form z = r(cosθ + i sinθ). Multiplying multiplies the moduli and adds the arguments — which is why z² has modulus r² and argument 2θ (De Moivre). The conjugate reflects z in the real axis.
1 Sequences & series AA & AI · SL & HL
Arithmetic sequences add a common difference; geometric ones multiply by a common ratio. Watch the terms — and their running sum — build.
nth term
Sum Sₙ
Examiner's note
Arithmetic: uₙ = a + (n−1)d, Sₙ = n/2(2a + (n−1)d). Geometric: uₙ = ar^(n−1), Sₙ = a(rⁿ−1)/(r−1). A geometric series converges only when |r| < 1 — the condition for a sum to infinity.
A.4 Rotational dynamics HL
The rotational analogue of Newton's second law: torque produces angular acceleration, resisted by the moment of inertia.
Torque τ
Moment of inertia I
Ang. accel. α
Examiner's note
τ = Iα mirrors F = ma. Torque is force × perpendicular distance, and I depends on how mass is distributed (for a uniform disc, I = ½mr²). More torque spins it up faster; more inertia resists — the same reasoning as linear motion, rotated.
A.5 Special relativity HL
Near the speed of light, moving clocks run slow and moving lengths contract. The Lorentz factor γ sets the scale of both.
Lorentz factor γ
Time (proper 1 s)
Length (proper 1 m)
Examiner's note
γ = 1/√(1 − v²/c²). A moving clock runs slow by Δt = γΔt₀ (time dilation); a moving length shrinks by L = L₀/γ (length contraction). Both are negligible at everyday speeds and dramatic near c. Keep straight which observer measures the proper quantity.
D.4 Electromagnetic induction HL
A magnet moving through a coil changes the flux, inducing an EMF. Faster motion, more turns, or a stronger magnet all raise the EMF.
Induced EMF (peak)
Examiner's note
Faraday: EMF = −N (dΦ/dt). The minus sign is Lenz's law — the induced current opposes the change that made it. No change in flux, no EMF: a magnet sitting still in the coil induces nothing, however strong.
D.3 Charged particle in a magnetic field SL & HL
A magnetic force perpendicular to the velocity bends the path into a circle. Its radius grows with momentum and shrinks with field.
Radius r = mv/qB
Period T
Examiner's note
The force F = qvB is always perpendicular to v, so it does no work — speed is constant, only direction changes. This gives circular motion with r = mv/(qB). Note the period is independent of speed — the basis of the cyclotron.
C.3 Double-slit interference SL & HL
Two coherent sources produce bright and dark fringes where their path difference is a whole or half wavelength.
Fringe spacing ∝ λ/d
Examiner's note
Constructive interference where the path difference is nλ; destructive at (n+½)λ. Fringe spacing s = λD/d — wider slits give narrower fringes. The pattern is direct evidence of the wave nature of light.
3 Vectors & the dot product AA HL & AI HL
Drag the two vectors. Their sum follows the parallelogram; the dot product and the angle between them update live.
a · b
Angle between
|a| , |b|
Examiner's note
a · b = |a||b|cosθ = a₁b₁ + a₂b₂. The dot product is zero exactly when the vectors are perpendicular — a fast test for orthogonality, and the route to the angle between two lines or planes in HL.
5 Maclaurin series AA HL
A polynomial built from derivatives at zero. Add terms one by one and watch it hug the curve over a widening interval.
Examiner's note
The Maclaurin series is f(0) + f′(0)x + f″(0)x²/2! + …. Each extra term improves the fit near x = 0 and extends the range where it is accurate. For sin and cos the series contain only odd or only even powers — a pattern worth recognising.
1 Roots of a complex number AA HL
The n roots of a complex number sit equally spaced on a circle — a regular polygon in the Argand plane.
Examiner's note
By De Moivre, the n roots of R∠φ are R^(1/n) ∠ (φ + 360°k)/n for k = 0…n−1. They share the same modulus and are spaced 360°/n apart — which is why they form a regular polygon. The nth roots of unity are the special case R = 1, φ = 0.
5 Slope fields AA HL & AI HL
A differential equation gives the gradient at every point. Drag a starting point and follow the solution curve through the field.
Examiner's note
A slope field shows the gradient dy/dx at each point without solving the equation. A solution curve follows those slopes — here traced by Euler's method. Different starting points give a family of solutions differing by the constant of integration.
5 Volume of revolution AA HL
Rotate the region under a curve about the x-axis and it sweeps out a solid. Stack thin discs to find its volume.
Disc estimate
Exact volume
Examiner's note
Rotating y = f(x) about the x-axis gives V = π ∫ₐᵇ y² dx — each thin disc has area πy² and thickness dx. Squaring the function before integrating is the step students most often miss. More discs, closer to the true volume.
A.2 Momentum & collisions SL & HL
Two objects collide on a frictionless track. Total momentum is always conserved; kinetic energy is only conserved if the collision is elastic.
Momentum (before = after)
KE before
KE after
Examiner's note
Momentum p = mv is a vector and is always conserved in a collision with no external force. In an elastic collision kinetic energy is also conserved; in an inelastic one, some becomes heat and sound. State which quantities are conserved before you calculate.
C.2 Refraction & Snell's law SL & HL
Light bends as it crosses between media. Beyond the critical angle it cannot escape — total internal reflection.
Refraction angle
Critical angle
Examiner's note
n₁ sinθ₁ = n₂ sinθ₂. Light entering a denser medium bends towards the normal. Total internal reflection occurs only going from dense to less dense, beyond the critical angle sinθc = n₂/n₁ — the principle behind optical fibres.
C.4 Resonance HL
A driven oscillator responds most strongly when driven near its natural frequency. Damping lowers and broadens the peak.
Response amplitude
Examiner's note
Resonance occurs when the driving frequency matches the natural frequency, giving maximum amplitude. Light damping gives a tall, sharp peak; heavy damping gives a low, broad one. It is why bridges and buildings are engineered to avoid resonant frequencies.
D.1 Orbital motion SL & HL
A planet on an elliptical orbit sweeps equal areas in equal times — moving fastest at its closest approach.
Speed (relative)
Examiner's note
Kepler's second law: the line from star to planet sweeps equal areas in equal times, so the planet speeds up at perihelion and slows at aphelion. This follows from the conservation of angular momentum — a link examiners like to draw out at HL.
1 Matrix transformations AI HL
A 2×2 matrix moves every point in the plane. The unit square shows how; the determinant is the area scale factor.
Determinant (area factor)
Examiner's note
The columns of the matrix are where the basis vectors î and ĵ land. The determinant ad − bc is the factor by which area is scaled — negative means the plane is flipped, zero means it collapses to a line.
3 Sine & cosine rule AA & AI · SL & HL
Drag the corners of the triangle. The side lengths and angles update — and tell you which rule to reach for.
Sides a, b, c
Angles A, B, C
Examiner's note
Use the sine rule a/sinA = b/sinB when you have a matching side–angle pair; the cosine rule a² = b² + c² − 2bc·cosA when you have two sides and the included angle, or all three sides. Choosing the right one first saves time.
5 Optimisation AA & AI · SL & HL
A rectangle sits under the parabola. Its area rises, peaks, then falls — calculus finds the exact width that maximises it.
Area
Maximum area
Examiner's note
To optimise, write the quantity as a function of one variable, differentiate, and set the derivative to zero. Here A(x) = 2x(4 − x²); solving A′(x) = 0 gives the exact maximising width. Always check it is a maximum, not a minimum.
Understanding the concept is the start. Turning it into marks is the work we do together.
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