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Which formula can be right?

The pendulum's period: √(L/g) or √(g/L)? Units decide. Pick the one formula whose kilograms, metres and seconds come out right.

  • Quantitative reasoning
  • Untimed
  • Endless levels

How it plays

A formula for a time must come out in seconds, a speed in metres per second, an energy in kg·m²/s². Each question gives the quantities and their units; only one option has the units the answer needs.

Levels go from two quantities to three or four — orbits, pressure, waves — then to finding the power a quantity must carry. Some questions offer a quantity that does not belong, like the mass of a pendulum bob.

Every option is checked by adding up the powers of kg, m and s, so exactly one is right by construction. Units cannot see pure numbers: √(L/g) is right in kind, and the working gives the true constant, here 2π. The wrong options are the real slips: the fraction upside down, the root forgotten, two powers swapped.

Keys

Space starts and skips ahead · 1–9 pick · Enter submits · Esc stops. On a phone, tap.

What it trains

This makes checking formulas by their units quicker and surer. It claims nothing wider.

Learn

How to get better at which formula can be right?: the method, the techniques, what each new level asks of you, and the traps the wrong answers are made of.

Your approach to each question

  1. Read the target’s units in the question — a time is s, a speed m/s, an energy kg·m²/s² — and keep them in view. Every option is judged against that one line.
  2. Turn each given quantity into powers of kg, m and s. The prompt prints them: g is m/s², k is kg/s², G is m³/(kg·s²).
  3. Check one unit at a time, the scarcest first. Usually that is kg: if the target has no kg, the kg must cancel or be absent. Then s, then m.
  4. Rule options out as soon as one unit fails; you seldom need to check all three for the wrong ones. The one that survives every unit is the answer.

Techniques and tricks

  • Kilograms first

    In most questions only one or two of the given quantities carry kg, so mass is the quickest unit to settle. If only one given quantity carries kg and the target has none, that quantity cannot appear at all. That is how the bob’s mass drops out of a pendulum’s period.

    e.g. pendulum period from L, g and m: only m has kg, and a time has none, so m is out

  • Spot the pairs that travel together

    Some quantities only make sense in pairs. G has kg⁻¹ and M has kg, so GM is m³/s² with no mass left. Treat GM as one block in every orbit question.

    e.g. √(GM/R): m³/s² ÷ m = m²/s², and the root gives m/s — a speed

  • A ratio that gives s² needs a root

    L/g is m ÷ (m/s²) = s². A time from a length and an acceleration must therefore be a square root. Whenever a ratio lands on a squared unit, the answer is its root.

    e.g. L/g = s², so √(L/g) is a time and L/g is not

  • Flip the fraction, flip the units

    An option that is the right one upside down has exactly the inverse units. √(g/L) is 1/s, not s. If an option looks right but its units come out inverted, the fraction is upside down.

  • Find the power by matching each unit

    Write the answer as a product of powers, then match the powers of kg, m and s on each side. That gives one small equation per unit; solve the one with a single unknown first.

    e.g. orbit period = R^a G^b M^c: s: −2b = 1, so b = −½; kg: −b + c = 0, so c = −½; m: a + 3b = 0, so a = 3/2

  • Know the derived units cold

    Force is kg·m/s². Energy is force times length, kg·m²/s². Pressure is force per area, kg/(m·s²). Power is energy per second, kg·m²/s³. Build each from the one before.

    e.g. ρgh: kg/m³ × m/s² × m = kg/(m·s²), a pressure

As the levels climb

  • The kinds unlock in this order: formulas from two quantities; then three or more — orbits, pressure, waves on a string, power; then finding the power a quantity must carry. Four in ten questions come from the newest kind.
  • From the second step of the ladder on, a quantity that does not belong may be offered with the others — a mass for a falling time, a stretch for a spring’s period, a length for a wave’s speed. Expect it, and use the kilograms-first check to throw it out.
  • The questions themselves stop growing early; what climbs is the clock, which tightens by about 7% a step between new kinds and on every level after, down to half. At the tight end, check kg and s first, and stop the moment only one option is left.

Traps to avoid

  • The fraction upside down: √(g/L) for √(L/g). Its units are 1/s, the inverse of the time asked for.
  • The root forgotten: L/g for √(L/g). The units are s², the square of what is asked. Any squared answer unit means a root is missing.
  • Two powers swapped: m²v for mv², or k²x for kx². The letters are right, so it looks right; count the powers of kg — energy has exactly one.
  • The quantity that does not belong let in: a pendulum’s period with m in it. If nothing else can cancel its kg, it has to go.
  • Expecting the pure number to show: units cannot see ½ or 2π, so the right option is kx², not ½kx². The working gives the true constant afterwards.

How to practise

  • Use “Kinds of question” to drill “Find the power” on its own. Solving for the powers is the method underneath every pick, and it makes the picks fast.
  • Raise “Options shown” towards 8 so the swapped-power and one-power-flipped options appear alongside the upside-down one, and a glance at one unit is not enough.
  • Read the working after every question, right or wrong, for its last line. It gives the real constant — 2π, ½, √2 — that the units could not tell you.

Levels

It is untimed unless you turn the clock on. There is no last level: at least one dial turns without end. Your level is judged on your last ten answers, every ten: 90% moves you up, below 80% moves you down, which holds most people near four in five right.

All of it is in the settings beside Start — how many answers a judgement looks at and how often it recalibrates, the thresholds, the time, which questions, how many options, and whether you see each verdict at once or review them together at the judgement. You can also jump to any level.

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