Lesson 3 of 6 · Algebra, direct proportion & inverse squares
Ask what stays fixed in an equation
8 minute lesson · 3 checks along the way
Your goal for this lessonRearrange a simple equation and distinguish direct, inverse, and inverse-square relationships while keeping the physical assumptions explicit.
Follow a linked name to explore its archive card. “Back to your lesson” brings you to the same part of the reading.
Words with a dotted underline open a short definition right here in the lesson.
Let the symbols keep track of the quantities
An equation records a relationship more compactly than a paragraph. Suppose a hypothetical instrument operates at a constant electrical power P for a duration t. Its energy use is E = P × t. If power is measured in watts and time in hours, energy is in . A watt is a rate of energy transfer; a watt-hour is an amount of energy. Their similar names do not make them interchangeable.
Given 180 watt-hours of usable energy and a steady load of 60 watts, rearrange the equation to t = E/P. Dividing 180 Wh by 60 W gives three hours. This is an ideal budget with no additional load or conversion loss. The algebra alone cannot tell you whether the stated energy is actually usable, the load stays constant, or the spacecraft has thermal and operational constraints.
Check 1 of 3
10 pointsUse the reading above. You can try again without losing points.
Separate linear from directly proportional
At fixed power, energy used is directly proportional to elapsed time: double the duration and double the energy. At fixed energy, operating time is inversely proportional to power: double the load and halve the available duration. The phrase “at fixed” does essential work. If power and usable energy both change, you need their new values rather than a shortcut based on only one of them.
A straight-line relationship need not be . Imagine a mission data file begins with a 20-megabyte header and adds 5 megabytes every minute. Its size is S = 20 MB + (5 MB/min) × t. At two minutes it is 30 MB; at four minutes it is 40 MB, not 60. The fixed starting term means that doubling elapsed time does not double the whole file.
Compare the evidence
A straight line is not always direct proportion
Compare two fictional file-generation models. Both have a constant rate, but only one starts at zero.
No starting file
- When and where
- Original fictional practice example · Training calculation; not mission telemetry
- Purpose
- Model only the generated data.
- What it supports
- S = (5 MB/min) × t: 10 MB at 2 min, 20 MB at 4 min.
- What remains uncertain
- This model deliberately omits any header or initial stored data.
A fixed header
- When and where
- Original fictional practice example · Training calculation; not mission telemetry
- Purpose
- Include a 20 MB starting contribution.
- What it supports
- S = 20 MB + (5 MB/min) × t: 30 MB at 2 min, 40 MB at 4 min.
- What remains uncertain
- The constant generation rate does not make the entire file directly proportional to elapsed time.
Which whole-file model doubles when elapsed time doubles from two to four minutes?
Compare your reasoning
Only the model that starts at zero doubles. In the second model, the newly generated part doubles from 10 to 20 MB, while the 20 MB header stays fixed. The final total grows from 30 to 40 MB.
Work through a decision
Use a ratio to test an inverse square
For a steady source spreading light approximately uniformly into unobstructed space, received power per unit area falls with the square of distance. Compare two ranges using F₂/F₁ = (r₁/r₂)². This avoids inventing a new constant. With the same source, a hypothetical of 100 W/m² at two astronomical units becomes 25 W/m² at four astronomical units: doubling distance leaves one quarter.
A mistaken estimate of 50 W/m² uses an inverse relationship without the square. Check the geometry: the same outgoing power spreads over a sphere whose area grows as distance squared. The comparison also assumes unchanged source output and no intervening absorption. It does not describe every extended nearby light source, and it is not a prediction of a solar panel’s complete electrical behavior.
Check 2 of 3
10 pointsUse the reading above. You can try again without losing points.
Check the factors outside the simple model
A panel’s collected power also depends on area, orientation, and efficiency. Doubling effective collecting area can double output only when the other relevant conditions remain fixed. BepiColombo’s documented solar-array rotation illustrates why orientation is an operational choice: pointing partly away from the Sun can reduce received power and help manage heating and cell degradation. A single photo does not determine the available watts.
When applying an equation, identify the unknown, isolate it, check units, and then test whether the result makes sense at an easy limiting case. Zero usable energy should not yield hours of operation at a positive load. A closer steady point-like light source should not become dimmer under the stated unobstructed model. These checks help locate mistakes while leaving room for real conditions that require a better model.
BepiColombo’s MTM monitoring camera recorded this solar-array view during the October 27, 2018 rotation. ESA explains that off-pointing changes collected power and helps manage heating and solar-cell degradation. This is an engineering camera observation, not a measurement of the hypothetical irradiance values in the lesson.
Look closely
Why does knowing the visible array’s area alone not determine its electrical output?
See what to notice
The illumination, angle to sunlight, efficiency, and operating conditions also matter. The perspective view does not directly measure area or establish those conditions. The lesson’s inverse-square comparison holds specified factors fixed.
Check 3 of 3
10 pointsUse the reading above. You can try again without losing points.
Put it together
Your lesson progress
Each check connects to a part of the story. Complete all 3 to earn your 25-point lesson bonus.
Go to the source
Explore the evidence behind this lesson. These optional readings and videos open in a new tab.