Dimensional Analysis, Scaling and the Order-of-Magnitude Estimate · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Let the units check your work

Science · Scientific Inquiry · ages 22-23
Name ______________________   Date ____________

  1. Speed is always a length divided by a time. Starting from one half m v squared, what are the dimensions of energy?

    • Mass times length squared over time squared
    • Mass over length times time
    • Length cubed over time

  2. Which of these additions is dimensionally allowed?

    • A length plus a time
    • A mass plus a speed
    • A length plus a length

  3. Both sides of a valid equation must carry matching dimensions.

    Circle one:   True   False

  4. You face a Fermi question like how many houses a proposed power plant could serve. What is your first move?

    • Guess the final number in one leap
    • Split it into guessable pieces and state each assumption
    • Look up every digit before thinking

  5. Your algebra for an energy ends with a term that adds 3 meters to 2 seconds. What do you conclude?

    • The calculation went wrong somewhere upstream
    • The answer is unusually precise
    • The units will cancel in the next line

  6. A chained conversion ends with seconds where you expected meters. What is the most likely cause?

    • A flipped conversion fraction
    • A rounding slip in the last digit
    • A wrong metric prefix

  7. A hall holds about 30 rows with about 40 seats each. Estimate the seat count to the nearest power of ten and type it.

    Answer: ______________

  8. Two careful Fermi estimates of one quantity give 300 and 4000. What do you say?

    • Both are fine, since estimates never disagree
    • At least one set of assumptions is off, because the answers span more than an order of magnitude
    • The larger one is automatically right
LightMySky · lightmysky.comW1-mt_jXnXRMKnC3-s1

Answer key

For grown-ups. Fold this page away before handing over the rest.

Let the units check your work W1-mt_jXnXRMKnC3-s1

  1. Mass times length squared over time squared · Squaring a length over a time and multiplying by mass gives mass times length squared over time squared.
  2. A length plus a length · You can only add things with the same dimensions.
  3. True · Mismatched sides mean the equation cannot describe one physical quantity.
  4. Split it into guessable pieces and state each assumption · Estimation means splitting the question, guessing each piece honestly, and multiplying.
  5. The calculation went wrong somewhere upstream · A sum of mismatched dimensions is meaningless, so no later step can rescue it.
  6. A flipped conversion fraction · Leftover wrong units mean a fraction equal to one was written upside down.
  7. 1000 · 30 times 40 is about 1200, whose nearest power of ten is 1000.
  8. At least one set of assumptions is off, because the answers span more than an order of magnitude · An order of magnitude answer should land near one power of ten, so a factor over ten apart means a bad assumption.
Worksheet · LightMySky