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.
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Dimensions, scaling and estimates

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

  1. Using kinetic energy = one half × mass × speed squared, what are the dimensions of energy?

    • Mass times length squared over time squared
    • Mass squared times length squared over time squared
    • Mass times length cubed divided by time squared

  2. Which addition passes the dimensional check?

    • A length added to a duration
    • A mass added to an object's speed
    • A length plus a length

  3. Both sides of a valid physical equation must have matching dimensions, although matching dimensions alone do not prove it correct.

    Circle one:   True   False

  4. Available average plant power is P and average demand per house is p. Which model estimates the number of houses supplied, assuming the stated power is available to them?

    • Multiply P by p to obtain a house count
    • Divide P by p to obtain a house count
    • Add P and p to obtain a house count

  5. A model fixes vt/L at a constant, where v is speed, t is time and L is distance. At fixed L, what happens to t when v doubles?

    • vt/L is dimensionless; t halves
    • vt/L has length units; t doubles
    • vt/L is dimensionless; t doubles

  6. To convert 4 m to cm, a student writes 4 m × (1 m / 100 cm), leaving m²/cm. Which diagnosis applies to this written setup?

    • The conversion fraction is upside down
    • The factor 100 should instead be 1000
    • Rounding changed the remaining units

  7. A hall has about 30 rows with about 40 seats each. Estimate its seat count, then give the nearest power of ten on the lesson's multiplicative scale.

    Answer: ______________

  8. Two estimates of one quantity are 300 and 4000. What can their ratio establish without knowing the true value?

    • At least one estimate is certainly invalid
    • They differ by over ten; investigate further
    • The larger estimate is certainly more accurate
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Answer key

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

Dimensions, scaling and estimates W1-mt_jXnXRMKnC3-s1

  1. Mass times length squared over time squared · Speed contributes length/time. Squaring and multiplying by mass gives mass × length²/time².
  2. A length plus a length · The length pair has matching dimensions. The other pairs do not.
  3. True · Dimensional consistency is necessary. A wrong coefficient or physical assumption can still pass that check.
  4. Divide P by p to obtain a house count · Available power divided by power per house gives the estimated house count.
  5. vt/L is dimensionless; t halves · vt/L has no dimensions. Keeping it constant at fixed L requires time to vary inversely with speed.
  6. The conversion fraction is upside down · The factor must be 100 cm / 1 m so the original m cancels, giving 400 cm.
  7. 1000 · 30 × 40 = 1200. Between 1000 and 10000, the multiplicative boundary is about 3162. Since 1200 is below it, choose 1000.
  8. They differ by over ten; investigate further · 4000/300 is about 13.3. This flags a large difference but does not prove an estimate invalid or identify the more accurate one.
Worksheet · LightMySky