Five calculators that show their working.
A solver that only prints a number teaches nothing. Each of these five names the equation it used, marks what you gave it against what it found, draws the situation where a drawing helps, and says out loud where the marks are usually lost.
Free, offline, and running on the same constants table as units and dimensions — so a number here can never disagree with a printed row. Jump to — kinematics · lenses & mirrors · circuits · hydrogen levels · significant figures.
Kinematics and projectiles
Tool 01Any three of u, v, a, s, t determine the motion. This solves for the other two, names the equation it used for each, warns when the square root has a second root you were meant to notice, and draws the velocity–time line whose area is the displacement.
Fill any three boxes; leave two empty. Signs matter — pick a positive direction and stay with it.
Launch speed and angle, the height it leaves from, and g as a positive magnitude. No air resistance — which is why real ranges fall short.
▸ The theory in three lines
The five SUVAT relations are not five facts but one: v = u + at integrated once more, with t eliminated in the two forms that do not want it. Any three of u, v, a, s, t fix the other two, so the whole skill is choosing the equation that misses the quantity you neither know nor want.
They hold only while a is constant. A graph with a curve in it, a rocket losing mass, a spring, a drag force — none of them are SUVAT problems, and using these equations there is the single most common wasted answer in mechanics.
A projectile is two SUVAT problems sharing a clock: uniform velocity across, uniform acceleration down. They are joined only by t. Landing speed √(u2 + 2gh) is independent of the launch angle — energy does not care about direction.
Lenses and mirrors
Tool 02The mirror and thin-lens equations, solved in whichever direction you need, with the magnification, the nature of the image in words, and a diagram drawn to scale along the axis. The sign convention is stated on the page, because getting it wrong is the whole of the difficulty.
Give two of f, u, v and leave the third empty. Every distance is measured from the pole or optical centre and is positive along the direction the light was travelling — so a real object in front is negative.
The lens maker’s formula. n is the material relative to its surroundings; both radii follow the same sign rule as every other distance.
A mirror has no formula to speak of: f is half the radius, for every material and every colour.
▸ The convention, and why it is the whole battle
One rule generates every sign: measure from the pole or the optical centre, take the direction of the incident light as positive, and take heights above the axis as positive. A real object therefore has u < 0 — always, in every question. Nothing else needs memorising, and no case-by-case table is needed.
The two relations differ by one sign because reflected light comes back: 1/v − 1/u = 1/f for a thin lens, 1/v + 1/u = 1/f for a mirror. Magnification follows: m = v/u for the lens, m = −v/u for the mirror. A negative m means inverted, and for a real object that also means real.
A converging element is the only one that can put a real image on a screen, and only when the object is outside f. Inside f it gives a virtual, magnified, erect image — that is a magnifying glass. A diverging element gives virtual, diminished and erect for every real object; if your answer says otherwise, a sign is wrong, not the physics.
Circuit reducer
Tool 03Type a network as an expression and watch it fold one combination at a time — then, with a source across it, the voltage, current and power in every single element. Capacitors and inductors too, with their rules the right way round. And a Wheatstone bridge solved by node analysis, balanced or not.
Write the network: + joins in series, | in parallel, brackets group, and | binds tighter — so 100|100+47 means (100∥100) then 47. Suffixes p n u m k M work: 2u is 2 μF.
Four arms P, Q, R, S with the galvanometer G across the middle and the cell across the ends. Solved by node equations, so it works balanced or not.
▸ The theory in three lines
Series means one path, so the current is shared and the potential divides; parallel means one pair of nodes, so the voltage is shared and the current divides. Resistance adds in series and adds reciprocally in parallel — and a parallel combination is always smaller than its smallest member, which is the fastest sanity check there is.
Capacitors do the opposite, and for a reason worth saying: in series the plates between the capacitors are isolated, so every capacitor carries the same charge, and 1/C adds. The smallest capacitor then takes the largest share of the voltage — and fails first.
Some networks fold and some do not. An unbalanced bridge has no series or parallel pair anywhere in it, and no amount of redrawing will produce one; that is precisely when you write node or mesh equations. Balanced, though — P/Q = R/S — the middle arm carries nothing and may simply be erased.
Hydrogen levels and spectra
Tool 04The Bohr model for any one-electron ion: level energies, the transition, its wavelength, frequency and wavenumber, the series it belongs to, and the orbit radius, speed and angular momentum. The level diagram is drawn to scale and the arrow takes the colour of the light actually emitted.
Z is the nuclear charge: 1 for hydrogen, 2 for He+, 3 for Li2+. The Bohr model is exact only for one electron.
▸ The theory in three lines
Bohr’s one postulate is that angular momentum comes in units of ℏ: mvr = nℏ. Everything else follows from Coulomb attraction and circular motion — r ∝ n2/Z, v ∝ Z/n, and Eₙ = −13.6 Z2/n2 eV. The energy is negative because the electron is bound; zero energy is a free electron at rest.
A photon is emitted or absorbed only for the difference between two levels, so a hot gas shows lines and not a continuum. Because the levels crowd together as n grows, each series bunches up towards its limit — and the series is named by the LOWER level: n = 1 Lyman, 2 Balmer, 3 Paschen.
The model gets hydrogen’s wavelengths right and almost everything else wrong: no fine structure, no intensities, and nothing at all for two electrons. Keep it as the one place where a quantised answer can be derived on paper in three lines.
Significant figures
Tool 05How many figures a number really claims, what it should be rounded to, and why 1500 is a badly written measurement. Then a drill — counting, rounding, products, sums and quoting a result against its uncertainty — marked with the reasoning, because the reasoning is the examinable part.
Type a measurement as you would write it — the count depends on how it is written, which is the whole point.
▸ The theory in three lines
Significant figures start at the first non-zero digit. Leading zeros are placeholders and never count; trailing zeros after a decimal point always do, because nobody writes them by accident. 1500 is ambiguous and 1.50×103 is not — scientific form exists to settle exactly this.
The two arithmetic rules are different, and swapping them is the classic loss: × and ÷ follow significant figures (the weakest factor wins), while + and − follow decimal places (the coarsest measurement wins). 12.3 + 0.4567 is 12.8, not 12.7567.
Round once, at the end. Carrying the full display through the working and quoting at the last step is not fussiness — rounding at every line is how a correct method produces an answer that disagrees with the key in the last figure.
Five kinds of question, at random
Counting figures, rounding, a product, a sum, and quoting a value against its error. Every answer comes back with the reason, and the score keeps your streak.
Three more calculators live on the units page — a converter across 30 families, a dimensional-homogeneity checker and an error-propagation tool. The mistakes these five are built around are indexed in concept traps; a term you want defined rather than computed is in the A–Z glossary.