Most physics questions in a school course have the same shape: a relationship between a handful of quantities, all but one of them given. The work is deciding which relationship applies, which is a question about what is constant, and then keeping the units straight while you use it. Both of those are places a good student loses marks and neither is difficult once it is named. This guide walks through the relationships in the order a course introduces them, states what each quantity means rather than only how it is computed, and points at the calculator for each one. Every tool runs in your browser, ignores air resistance unless it says otherwise, and shows the formula with your numbers substituted rather than only the answer.
The shape of almost every question: one relationship, one unknown
Speed, distance and time related by s = d ÷ t is the smallest complete example. Give two of the three and the third is determined; the interesting part of the problem is never the division, it is knowing that the division is what applies and that the distance is a total rather than a displacement. Every calculator here works the same way: enter the quantities you know, leave the one you want blank, and the page solves for it — which is why the number fields accept an empty box as a meaningful state rather than as a zero. There is no submit button, because a form that computes on every keystroke lets you see how the answer moves as you change an input, which is the most direct way to understand which quantity controls what. Where a relationship has two arrangements that are genuinely different statements — power as work over time against power as force times velocity, for example — the page offers them as a choice rather than hiding the second behind a fourth field.
Units are chosen per field, because that is how problems are written
A problem that gives a mass in grams, a speed in km/h and wants an answer in kilojoules is not unusual, and it is not a reason to convert by hand before you start. Each field here carries its own unit selector: metres, centimetres, kilometres, feet, miles; seconds, minutes, hours; joules, kilojoules, calories, kilowatt-hours. Every value is converted to SI on the way into the formula and converted back into the unit you chose for the answer, so the arithmetic stays in one system — metres, seconds, kilograms, radians — and the conversions happen where they belong, at the edges. The one convention worth stating because no page can enforce it: temperatures in gas problems must be absolute, so 25 °C is 298.15 K and not 25.
Kinematics: know which quantity is constant before you pick a formula
Speed is distance over time; velocity is displacement over time and carries a direction; acceleration is the change in velocity over the time it took. The distinction between distance and displacement is what makes a round trip have a non-zero average speed and a zero average velocity, and the distinction between average and instantaneous is why 100 m in 20 s is 5 m/s even if the runner started from a standstill and finished at 10. Constant acceleration is the case school problems almost always mean, and it is the case where a single pair of equations covers everything — which is exactly what free fall is, with a fixed downward acceleration. Two questions get asked about a falling object and they are not the same question: how long does it take to fall this far, and how far has it fallen after this long. Given a height of 20 m and g = 9.8 m/s², the first is t = √(2h ÷ g) = 2.02 s; given two seconds instead, the second is h = ½gt² = 19.6 m. Both ignore air resistance, which makes them an approximation that is excellent for a stone and poor for a feather.
Projectiles: two one-dimensional problems sharing a clock
The trick with projectile motion is that the horizontal and vertical parts are independent: nothing you do horizontally changes how fast you fall. So the launch velocity is decomposed into a horizontal part that stays constant (vx = v₀ cos θ) and a vertical part that changes under gravity (vy = v₀ sin θ − gt), and the flight ends when the vertical position returns to the ground. That last step is where the launch height matters. From a rooftop, the time of flight is not 2vy/g — it is the positive root of a quadratic, because the object has to fall past its starting height. That is why the calculator asks for h₀ as a required input rather than defaulting it: a throw from 10 m genuinely travels further than the same throw from the ground, and a tool that assumed otherwise would be wrong in a way that is hard to notice. The plot and the time-step table on the page come from the same sampled trajectory, so they cannot disagree with each other, and both ignore air resistance — a real throw lands slightly short.
Forces, energy and momentum: three bookkeeping systems, each with its own conservation
Newton's second law, F = ma, is the bridge between a force and the motion it causes, and it is worth remembering that one newton is the force that accelerates one kilogram at one metre per second squared. Work is force applied along a distance, W = Fd cos θ, where θ is the angle between the force and the direction of travel — so a force at 90° to the motion does no work at all, which is why carrying a bag horizontally is tiring but not, in this sense, work. Energy comes in two forms worth distinguishing: kinetic, ½mv², which quadruples when you double the speed because the speed is squared, so a car at twice the limit carries four times the energy to lose; and gravitational potential, mgh, which is energy stored by lifting a mass against gravity and returned when it falls. Momentum, p = mv, is the quantity that is conserved whenever no external force acts, and that makes collisions a bookkeeping exercise rather than a mechanics problem: total momentum is identical before and after, in both an elastic collision (where kinetic energy also survives) and a perfectly inelastic one (where the objects move off together and kinetic energy is lost to deformation, sound and heat). Comparing the momentum and the kinetic energy before and after is what distinguishes the two, and it is why the collision page reports both.
Circuits: four quantities, two relations, and any two values are enough
Ohm's law, V = IR, relates voltage, current and resistance, and one more relation ties power into the same set: P = VI, which is also I²R and V²/R, depending on which two quantities you already have. That is why the electrical tools take any two of P, V, I and R and return all four with the arrangement they used shown underneath — the pairs are enumerated rather than inferred, because which form of the power equation applies depends entirely on the pair you started with. Resistance is where the intuition needs adjusting. In series, resistances simply add and the same current flows through each, so voltage divides in proportion to resistance. In parallel, the reciprocals add and the equivalent resistance is always *smaller* than the smallest resistor in the network — most of the current takes the path of least resistance — which is why the parallel solver uses the reciprocal form and reports each resistor's share of the current rather than its fraction of the total resistance.
Waves: when two values are not enough
A wave relates four quantities through two relations: v = fλ says the speed is the frequency times the wavelength, and T = 1/f says the period is the reciprocal of the frequency. Normally two of the four determine the rest, but frequency and period are the same information written two different ways, so giving only those two leaves both the speed and the wavelength undetermined — there is no third relation to pin them down. That case is reported rather than guessed at, because a page that invented a wavelength would give a confident and meaningless answer. Give frequency or period plus either the speed or the wavelength and everything follows, which is also why the wave tools sit next to the audio ones: 440 Hz is the A above middle C and 343 m/s is the speed of sound in air at about 20 °C, so the wavelength of that note comes out at roughly 0.78 m — the number an audio tool needs when it is converting between time and frequency.
Fluids and the sanity check that catches most mistakes
Density is mass per unit volume, ρ = m/V, and pressure is force spread over an area, P = F/A — the same force on a smaller area gives a larger pressure, which is the whole reason a sharp knife cuts. Both are relationships where the useful habit is comparing the answer to a landmark rather than reading it as a bare number. Water is 1000 kg/m³, which is also 1 g/cm³, so anything that comes out an order of magnitude away from that for a familiar material deserves a second look, and a pressure near 100 kPa is roughly atmospheric. Landmarks catch the two errors that unit conversion cannot: a volume entered in cm³ where litres were meant, and an area entered in cm² where m² were meant. Finally, report the answer to the precision the data supports — round once, at the end — and prefer the substituted formula the tool prints, since a result you can retrace is one you can defend.