What’s New with Step-by-Step Physics and More!

Jason Martinez, Wolfram|Alpha LLC

#WolframTechConf

ABSTRACT

In the past few years, we’ve added hundreds of new formulas to Wolfram|Alpha and greatly expanded step-by-step support. Come learn about the expansion of step-by-step coverage to a new range of empirical formulas, thermodynamic calculations and equations that involve temperature values. We will be showing off new step-by-step derivations, including new diagrams and formula descriptions to provide better physical understanding of problems. Also see Wolfram|Alpha’s new handling of nonphysical results to inform users of the calculators when their inputs lead to values outside the physical domain.

Formula Implementation Over Time

Over the past few years, hundred of formulas have been added to Wolfram|Alpha and exposed via FormulaData. At the same time we’ve expanded the percentage of formulas covered by our Step-by-Step functionality.

(*date, Wolfram|Alpha version, count*)
waTimeSeries={{{2009,5,18},430, "WAB_1"},{{2009,9,18},550, "WAB_19"},{{2010,1,18},689, "1.0.34"},{{2010,5,3},710, "1.0.50"},{{2011,3,30},831, "1.0.97"},{{2011,11,4},860, "1.1.2"},{{2012,4,2},1027, "1.1.50"},(*{{2013,3,13},1165, "1.1.95"},*){{2013,2,27},1184, "1.2.1"},{{2014,1,27},1228, "1.2.50"},{{2015,1,12},1259, "1.2.100"},{{2016,1,12},1274, "1.2.152"},{{2017,1,11},1283, "1.2.204"},{{2018,1,18},1285, "1.2.256"},{{2018,2,5},1285, "1.3.2"},{{2018,7,26},1291, "1.3.26"},{{2019,1,3},1293, "1.3.50"},{{2019,6,24},1294, "1.3.74"},{{2019,12,13},1301, "1.3.98"},{{2020,5,29},1311, "1.3.122"},{{2020,11,4},1329, "1.3.146"},{{2021,2,4},1331, "1.4.12"},{{2021,5,13},1335, "1.4.24"},{{2021,10,27},1338, "1.4.48"},{{2022,1,28},1345, "1.4.60"},{{2022,5,26},1345, "1.4.72"},{{2022,7,22},1338, "1.4.84"},{{2022,10,13},1342, "1.4.96"},{{2023,1,5},1349, "1.4.108"},{{2023,4,3},1377, "1.4.120"},{{2023,6,22},1389, "1.4.132"},{{2023,9,11},1409, "1.4.144"},{{2023,12,7},1412, "1.4.156"},{{2024,3,4},1441, "1.4.168"},{{2024,5,16}(**),1449, "1.4.180"},{{2024,7,29},1480, "1.4.192"},{{2024,11,12},1492, "1.4.204"},{{2025,2,11},1979, "1.4.216"},{{2025,4,28},2007, "1.4.228"},{{2025,7,41},2111, "1.4.240"},{{2025,10,13},2168, "1.4.260"}};
DateListPlot[waTimeSeries[[All,;;2]],Joined->True,PlotRange->All,PlotLabel->"Wolfram|Alpha formulas"]

(*date, FormulaData paclet version/Wolfram Language version, count*)
fdTimeSeries={{{2014,10,6},"1.0.1/10.0", 1072}, {{2015,2,15},"1.0.1/10.1", 1081}, {{2015,7,22},"1.0.1/10.3", 1083}, {{2016,4,6},"1.0.1/10.4", 1085}, {{2017,1,2},"1.0.1/11.0", 1100},{{2017,9,26},"1.0.1/11.3", 1102}, {{2019,3,15},"1.1.1/12.0", 1118}, {{2019,6,24},"1.1.1/12.1", 1120},{{2020,9,29},"1.1.3/12.2", 1161},{{2021,3,15},"1.1.9/12.3", 1177},{{2021,9,27},"1.1.15/13.0", 1200},{{2022,3,24},"1.1.17/13.1", 1219},{{2022,10,3},"1.2.3/13.2", 1226},{{2023,3,28},"1.3.3/13.3", 1268},{{2023,5,2},"1.3.4/14.0", 1268},{{2024,5,11},"2.0.0/14.1", 1422},{{2024,10,15},"2.0.2/14.2", 1489},{{2024,12,5},"2.0.5/14.3", 2008}, {{2025,7,2},"2.0.9/14.3+", 2272}};
DateListPlot[fdTimeSeries[[All,{1,3}]],Joined->True,PlotRange->All,PlotLabel->"FormulaData entities", Epilog->{Orange,Tooltip[Point[{AbsoluteTime[{2014,10,6}],1072}],"10.0"], Tooltip[Point[{AbsoluteTime[{2017,1,2}],1100}],"11.0"],Tooltip[Point[{AbsoluteTime[{2019,3,15}],1118}],"12.0"],Tooltip[Point[{AbsoluteTime[{2021,9,27}],1200}],"13.0"],Tooltip[Point[{AbsoluteTime[{2023,5,2}],1268}],"14.0"],Tooltip[Point[{AbsoluteTime[{2024,12,5}],2008}],"14.3"]}]

Sample formulas

Here are just a small sample of the formulas added in the past year:

WordCloud[{Log[21]->"creatinine clearance",Log[15]->"fluid replacement requirement",Log[20]->"negative likelihood ratio",Log[27]->"false positive rate",Log[12]->"characteristic acoustic impedance",Log[520]->"second dodecahedron stellation surface area",Log[7]->"Air Quality Health Index (Canada)",Log[8]->"mammal growth rate",Log[2]->"airplane pressurization",Log[7]->"propane tank filling calculator",Log[75]->"Jain equation",Log[2]->"aquarium glass thickness",Log[22]->"dissipation factor of capacitor",Log[12]->"ideal half Atwood machine on an inclined plane",Log[16]->"ball mill critical speed",Log[4]->"bend deduction",Log[4]->"blood pH",Log[8]->"terminal velocity of rising bubbles",Log[9]->"Janka hardness",Log[6]->"electroplating",Log[2]->"cetane index by four variable equation",Log[3]->"Shannon–Hartley theorem",Log[3]->"Clapeyron equation",Log[5]->"thermodynamic beta",Log[11]->"Sellmeier equation",Log[3]->"coefficient of system performance",Log[4]->"coherence length of two waves",Log[17]->"plasma frequency",Log[7]->"push\[Hyphen]up fitness test",Log[3]->"Debye frequency",Log[2]->"diffusion distance",Log[3]->"distance below the horizon",Log[4]->"Yule\[Hyphen]Nielsen formula",Log[4]->"Einstein ring",Log[4]->"stellar hydrogen burning rate",Log[2]->"energy to launch",Log[7]->"floor\[Hyphen]to\[Hyphen]floor performance time",Log[6]->"grade horsepower",Log[5]->"simplified net tonnage (USCG)",Log[3]->"Kimura model",Log[19]->"propulsion resistance for freight rolling stock - French locomotives",Log[7]->"reproducibility—appraiser variation",Log[2]->"perception of time",Log[3]->"Shannon–Hartley theorem",Log[2]->"propagation delay",Log[3]->"tree net oxygen generation",Log[3]->"tree net oxygen generation",Log[2]->"volume fraction"}]

Step-by-Step Coverage

In addition to new formulas, Wolfram|Alpha has increased the fraction of formulas covered by Step-by-Step calculations. This has involved generating steps when hidden factors are present and handling temperature arithmetic.

Empirical Formulas & Hidden Factors

Many formulas, particularly empirical formulas, contain hidden factors that omitted to avoid cluttering the expression. 

To accurately providing steps, these factors have to be exposed.

Some examples include pH:

Other examples include BMI, or body mass index, and the Coriolis effect. For BMI, Wolfram|Alpha also must convert the hidden factor in such a way to make it consistent with the input units:

For the Coriolis effect, the hidden factor is a conversion between radians, an angular unit, and revolutions.

More Steps Coverage

A side effect of properly supporting steps for formulas with hidden factors is support for formulas that use hidden equations.

Dealing with Temperature

Temperature units pose a unique challenge since they exist along a scale which complicates any arithmetic involving them.

(Quantity[283, "K", kelvins, "Kelvins"]) - (Quantity[0, "°C", degrees Celsius, "DegreesCelsius"])

Temperature versus Temperature Difference

For temperature conversions we need to account for the scale. For example in the steps for the fluid temperature derived from the Brinkman number:

An example of temperature difference unit conversions can be found in the steps for boiling point elevation:

Temperature Conversions

Most formulas where temperature arithmetic is used assume you are working in Kelvins. In thermal deformation, for example, one of the first steps is to ensure that all input values transformed to the correct absolute temperature scale:

Unit System Specific formulas

Not all formulas assume a Kelvin scale however. Some empirical formulas like the calculation of Pasteur units or thermal degradation units assume that calculations will occur in degrees Celsius.

Nonphysical Quantities

Wolfram|Alpha has improved its recognition of what values lie outside the physically acceptable range and described how the results of a formula exceed those limits.

What is Nonphysical?

Nonphysical quantities are those can’t be realized in reality. Examples include complex values (like 1+i ft), negative values for positive only physical quantities (like negative mass or - 2 Kelvin) and values that go outside the possible range.

centripetal acceleration 2m/s, -1 m/s^2

equivalent sound level for fixed sources using steady sound level, steady sound level 300 dB

Exceptions

Due to the limitations of the Wolfram|Alpha interface some exceptions are made.

There isn’t an intuitive way to include vectors, so in one dimensional formulas Wolfram|Alpha uses negative values for velocity, acceleration and distance to indicate direction. Strictly speaking these should be vectors and negative values would be incorrect.

In other cases we leave the door open to nonphysical values since they represent some of the fun of working with Wolfram|Alpha. Examples include superluminal velocities or atomic number (for very massive if unlikely elements).

Improvements to Input Field checks

Wolfram|Alpha has always had limits on what would be considered a valid input value for a formula (based on whether it was physically valid or within the regime where the formula itself was valid). Now it takes into consideration the physical limits from our unit framework and uses that to limit parsing and input fields.

moment of inertia of a cone calculator, mass

moment of inertia of a cone - 1 kg m^2

Nonphysical Step Results

The highlighting of nonphysical results does not disrupt the Step-by-Step functionality with explanations of the valid regimes incorporated into the steps themselves.

centripetal acceleration 2m/s, -1 m/s^2

equivalent sound level for fixed sources using steady sound level, steady sound level 300 dB

Step Improvements

Step-by-Step Quantity Simplification

To further expand Wolfram|Alpha formula steps, a Step-by-Step expansion of the simplification of the units of quantities will soon be released.

average velocity calculator speed=2 meters/second

Reynolds number formula

photon energy E=8.203*^-19 J

Derivations and Illustrations

Wolfram|Alpha includes more illustrations in steps to illustrate the systems in question and convey physical understanding.

projectile thrown straight up

projectile released from a height of 10 meters

height of a bouncing ball after 3 bounces

Force & Velocity Components: force components, Fx=10 N, F=20 N

Smaller Introductions

In addition to larger dedicated derivations and illustrations many formulas, particularly in electromagnetism, thermodynamics and statistical mechanics, have received single step introductions and graphics.

centripetal acceleration calculator

train velocities problem

Boyle's Law calculator

Graham's Law calculator

probability of exceedance

New Dedicated Steps Coverage

In addition to providing steps for these calculations, Wolfram|Alpha will include graphics to help understand the geometry of the problems.

Charged Balls

charged ball r=0.5m

Shadow Length

shadow length for a local light source

shadow length for shadows cast by the sun reference object's height

shadow length for a local light source shadow growth rate

shadow length for a local light source shadow tip speed

FormulaData & Around

FormulaData has been upgraded to deal with Around and Intervals in inputs, allowing formula calculations to be done with rigorous attention to uncertainty.

FormulaData["OhmsLaw",{"V"->Quantity[Around[1,0.02],"Volts"],"I"->Quantity[2,"Amperes"]}]

FormulaData["OhmsLaw",{"V"->Quantity[Around[1,0.05],"Volts"],"I"->Quantity[Around[2,0.01],"Amperes"]}]

FormulaData["OhmsLaw",{"V"->Quantity[Interval[{1,2}],"Volts"],"I"->Quantity[2,"Amperes"]}]

Final Remarks

As we continue to expand our formula coverage we also want to make it easier to find what formulas we have.

FormulaLookup["Kinematics"]//Shallow

If there is some formula or Step-by-Step content you would like to see let us know.