Named derived units

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Many physical quantities are really other quantities multiplied together. A force is a mass times an acceleration; power is voltage times current; energy is power times time, or a force times a distance. When you multiply two quantities, the engine tracks what their units combine into, and when the result is one of these named quantities, it shows it by its name:

70 kg * 9.81 m/s^2 as N // 686.70 N
230 V * 13 A as W // 2,990.00 W
50 N * 4 m as J // 200.00 J
2000 W * 3 hours as kWh // 6.00 kWh

N is the newton (force), W the watt (power), J the joule (energy), and kWh the kilowatt-hour (energy again, the unit an electricity bill uses). Writing as N asks for the answer in that unit; the engine also names the result on its own when you leave the as off. Dividing names a result the same way: an energy over a time is a power, and a force over an area is a pressure, in pascals (Pa).

100 J / 5 s // 20.00 W
200 N / 2 m² // 100.00 Pa
100 Pa * 2 m² // 200.00 N

Every spelling of a quantity takes part, imperial as well as metric, so pounds, feet and pound-force (lbf) compose just as kilograms, metres and newtons do.

10 lbf * 3 ft // 40.67 J
5 lb * 9.8 m/s^2 // 22.23 N

Dividing a power by a voltage gives the current, the rate at which electric charge flows, in amperes (A): a 100 watt load on 20 volts draws 5 amps. It converts to milliamps and kiloamps like any other unit.

100 W / 20 V // 5.00 A
100 W / 20 V in mA // 5,000.00 mA
2 A * 12 V // 24.00 W

A power used for a time of a minute or more is named in watt-hours, with the power’s own prefix: a 2 kW heater for three hours uses 6 kilowatt-hours, the figure a bill charges for, rather than 21,600,000 joules. Dividing such an energy by a time gives the power back in the matching watt. A power for seconds stays in joules, the energy’s own scale.

2 kW * 3 h // 6.00 kWh
100 W * 3 hours // 300.00 Wh
6 kWh / 3 h // 2.00 kW
100 W * 30 s // 3,000.00 J

The electrical quantities are named the same way. A voltage over a current is a resistance, in ohms (Ω); a voltage over a resistance, or a power over a voltage, is a current in amps (A); and a current for a time is a charge, named in amp-hours (Ah), the unit a battery is rated in. A charge in amp-hours at a voltage is the battery’s energy, named in watt-hours. Electricity walks through each.

12 V / 2 A // 6.00 Ω
12 V / 6 Ω // 2.00 A
2 A * 3 h // 6.00 Ah
3000 mAh * 3.7 V // 11.10 Wh

A prefix in front of a unit scales it by a power of ten: k (kilo) is a thousand, m (milli) a thousandth, M (mega) a million, µ (micro) a millionth. as takes any of them before the newton, joule, watt, watt-hour and pascal, and the millivolt and kilovolt beside the volt. The case of a prefix letter is part of its meaning, so mW (a milliwatt) and MW (a megawatt) are a billion times apart, and pW (pico) and PW (peta) further still. as reads the unit exactly as written, just as in does:

5 W as mW // 5,000.00 mW
5 W in mW // 5,000.00 mW
5 MW as mW // 5,000,000,000.00 mW
1 W as pW // 1,000,000,000,000.00 pW

The spellings as reads, from pico (a trillionth) to peta (a thousand million million):

unit	spellings
newton	pN, nN, µN, mN, N, kN, MN, GN, TN, PN
joule	pJ, nJ, µJ, mJ, J, kJ, MJ, GJ, TJ, PJ
watt	pW, nW, µW, mW, W, kW, MW, GW, TW, PW
watt-hour	pWh, nWh, µWh, mWh, Wh, kWh, MWh, GWh, TWh, PWh
pascal	pPa, nPa, µPa, mPa, Pa, kPa, MPa, GPa, TPa, PPa
volt	mV, V, kV

A spelling in the wrong case is refused rather than read as one of the two it could be. as mw names no unit, since it could be the milliwatt or the megawatt, and as MV is not the millivolt, whose prefix is the lower-case m. A letter that is not a prefix carries no such meaning, so as n is still the newton and as KWH the kilowatt-hour.

5 W as mw // ERROR: "as mw" could be "as mW" or "as MW": write the unit with its prefix in its own case (m is milli, M is mega, p is pico, P is peta)
1 V as MV // ERROR: "as MV" is not a unit: the one spelled alike is "as mV", whose prefix is written in the other case (m is milli, M is mega, p is pico, P is peta)
10 N as n // 10.00 N

Speed, acceleration and frequency

A speed is a distance per time, such as kilometres per hour. An acceleration is how fast a speed changes, a speed per time: a falling stone gains 9.81 metres per second every second, written 9.81 m/s^2, metres per second squared. A frequency is how often something happens per second, in hertz (Hz): 10 Hz is ten times a second.

These combine the way a physics book says they do. An acceleration for a time is the speed gained; a change of speed over a time is an acceleration; a speed times a force is a power; and a frequency for a time is a plain count of how many times it happened.

9.81 m/s^2 * 3 s // 29.43 m/s
3 s * 9.81 m/s^2 // 29.43 m/s
100 km/h / 10 s // 2.78 m/s²
29.43 m/s / 9.81 m/s^2 // 3.00 s
10 Hz * 2 s // 20
10 m/s * 5 N // 50.00 W

The answers convert like any other speed or acceleration. A speed is built in metres per second, or in the acceleration’s own length and time when it was written in feet or hours, and an acceleration in metres per second squared.

9.81 m/s^2 * 3 s in mph // 65.83 mph
3 ft/s^2 * 2 s // 6.00 ft/s
100 km/h / 10 s in ft/s^2 // 9.11 ft/s²

An acceleration can be written in any length over a time squared, ft/s^2 or km/h^2 as well as m/s^2, and they convert into one another. Anything else asked of one is refused in words: an acceleration is not a force until a mass multiplies it.

9.81 m/s² in m/s^2 // 9.81 m/s²
9.81 m/s^2 in ft/s^2 // 32.19 ft/s²
2 kg * 3 ft/s^2 // 1.83 N
9.81 m/s^2 in N // ERROR: an acceleration cannot be converted to a force

A frequency is a count per second, so it converts to any count per unit of time written with a bare slash, and back again.

10 Hz in /s // 10.00 /s
10 Hz in /min // 600.00 /min
600 /min in Hz // 10.00 Hz

A speed is only built this way when one side is already a speed, an acceleration or a frequency. A distance over a time keeps the units it was written in, 90 km / 3 days is a rate in kilometres per day, and a distance over a speed is a time in the speed’s own hours (see rates and speeds).

The boundary

This works only where the combination makes a named quantity, and multiplying two unrelated quantities is still reported as a mismatch rather than invented into a unit. Lengths are the exception that needs no name: a length times a length is an area and a length times an area is a volume, so 5 m * 3 m is 15.00 m². Those rules, and rates that cancel against what they are per, are on multiplying and dividing units.

The boundary: the named units are the newton, joule, watt, pascal, volt, ohm, ampere and amp-hour, with the speeds, accelerations and counts above. A combination with no name among them, such as a kilogram-metre or a momentum (100 kg * 10 m/s), is not shown as a compound unit, and neither is a time worked out from quantities other than an acceleration, so an energy over a power stays kWh/kW. Torque is not covered, and the mole is not part of this arithmetic at all (see moles).