Math Detours · Grade 6 · Sharpen · 17

The 125-million-dollar unit mistake

In 1999 a NASA spacecraft flew into Mars instead of around it, because two teams used different units and nobody checked. Tonight you learn the checking trick that would have saved it.

Name ____________________Date ____________

Tonight's idea

The Mars Climate Orbiter cost about 125 million dollars. The company that built it wrote software that reported small thruster pushes in pound-force seconds. NASA's navigation software assumed the numbers were in newton seconds. One pound-force is 4.448 newtons, so every push was recorded as about four and a half times weaker than it was. Over months of corrections the orbiter drifted, arrived 170 km too low, and broke up in the atmosphere.

The fix is a habit. Every quantity carries its unit, and units obey algebra. You can multiply and divide them, cancel them like fractions, and if the units of your answer come out wrong, the answer is wrong, no matter how careful the arithmetic was.

Converting is just multiplying by one. Since \(1000\ \text{m} = 1\ \text{km}\), the fraction \(\frac{1000\ \text{m}}{1\ \text{km}}\) equals 1, and so does its upside-down version. Multiply by whichever one cancels the unit you have and leaves the unit you want. Chain as many as you need.

And the rule nobody states: you can only add things with the same units. Three metres plus two seconds is not five of anything. If a formula asks you to add them, the formula is wrong.

Watch one: 60 miles per hour in metres per second

One mile is 1609 m. One hour is 3600 s.

\[ \frac{60\ \cancel{\text{mi}}}{1\ \cancel{\text{h}}} \times \frac{1609\ \text{m}}{1\ \cancel{\text{mi}}} \times \frac{1\ \cancel{\text{h}}}{3600\ \text{s}} = \frac{60 \times 1609}{3600}\ \frac{\text{m}}{\text{s}} \approx 26.8\ \text{m/s} \]

Watch the units: miles cancel, hours cancel, metres over seconds remain. If you had flipped the hour fraction the wrong way, the hours would not cancel and you would be left with metre-seconds per hour squared, not the metres per second you wanted. That is the signal to try again. Units catch most slips. Check the size of the answer too.

Your turn

  1. Convert 90 km/h to m/s. Show the fractions and the cancelling.
  2. One litre is 1000 cm³. A fish tank is 50 cm × 25 cm × 30 cm. How many litres does it hold?
  3. A recipe needs \(\tfrac{3}{4}\) cup of milk. One cup is about 240 mL. How many millilitres? How many litres?
  4. How many seconds are in a fortnight (14 days)? Write the chain of fractions.
  5. A wooden block has mass 2 kg and measures 20 cm × 10 cm × 12.5 cm. Water has density 1 g/cm³. Find the block's density in g/cm³ and decide whether it floats. (Be careful with kg and g.)
  6. The orbiter's software reported a push of 1 pound-force second, and NASA's software read it as 1 newton second. What was the real push in newton seconds? By what factor was NASA's model wrong about the size of every push, and was it too big or too small? Do the units alone tell you whether the orbit ended up too high or too low? (Think about what information they carry.)

Stretch: a light-year from scratch

Light travels 299,792 km every second. Using the number of seconds in a year (see the piano tuners sheet, or Big and tiny numbers), work out how many kilometres a light-year is. Write it in scientific notation. Then explain a second trap: \(1\ \text{m} = 100\ \text{cm}\), but \(1\ \text{m}^2\) is not \(100\ \text{cm}^2\). How many is it, and why?

Answers

  1. \(\frac{90\ \text{km}}{\text{h}} \times \frac{1000\ \text{m}}{\text{km}} \times \frac{1\ \text{h}}{3600\ \text{s}} = 25\ \text{m/s}\). (Dividing km/h by 3.6 is the shortcut.)
  2. \(50 \times 25 \times 30 = 37{,}500\ \text{cm}^3 = 37.5\) litres.
  3. \(0.75 \times 240 = 180\) mL, which is 0.18 L.
  4. \(14\ \text{days} \times \frac{24\ \text{h}}{\text{day}} \times \frac{60\ \text{min}}{\text{h}} \times \frac{60\ \text{s}}{\text{min}} = 1{,}209{,}600\) s.
  5. Volume \(20 \times 10 \times 12.5 = 2500\ \text{cm}^3\). Mass 2 kg = 2000 g. Density \(2000 / 2500 = 0.8\ \text{g/cm}^3\), less than water, so it floats (with 80% of it under the surface).
  6. Real push \(4.448\) newton seconds. NASA's model believed every push was 4.448 times smaller than it really was, so the spacecraft was being nudged more than predicted. The units tell you the size of the mistake, but not the direction of the orbit error: that depends on which way the thrusters were pointing and where the craft was. In the real event the accumulated error brought it in about 170 km too low.

Stretch. With 365.25 days in a year, \(299{,}792 \times 31{,}557{,}600 \approx 9.46 \times 10^{12}\) km (365 days gives 9.45), about nine and a half trillion kilometres. Area: a square metre is a square 100 cm on each side, so it contains \(100 \times 100 = 10{,}000\ \text{cm}^2\). Units of area square along with the lengths, and units of volume cube: \(1\ \text{m}^3 = 1{,}000{,}000\ \text{cm}^3\).

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