Every number below is off our kart (the kart's setup record, the 9/19 session entry, our speed calculator).
The one idea
A physical quantity is a number AND a unit. "12" means nothing; "12 psi" means something. When you multiply or divide quantities, the units multiply and divide too, and they have to come out as the unit you wanted. If they don't, the reasoning is wrong, no matter who said it.
That check is called dimensional analysis. It's the cheapest sanity test in engineering, and it works on setup arguments as well as equations.
Three times skipping the check was expensive
- Mars Climate Orbiter, 1999. One team's software output thruster impulse in pound-force-seconds; the other team's software read it as newton-seconds. The numbers were 4.45x off. The spacecraft entered the Martian atmosphere 100 km too low and burned up. $327 million.
- The Gimli Glider, 1983. An Air Canada 767 was fueled in pounds when the paperwork needed kilograms (Canada had just gone metric). It took off with under half the fuel. Both engines quit at 41,000 ft and the captain glided it to a dead landing on a closed airstrip that was hosting a drag race that day. Nobody died.
- The kilogram used to be a lump of metal in a vault in Paris. Every scale on Earth was, in the end, calibrated against that one object. It was slowly losing mass (about 50 micrograms in a century) and nobody could say whether IT was changing or everything else was. In 2019 the kilogram was redefined in terms of a constant of nature (Planck's), so it can never drift again.
The table you actually need
| From | To | Multiply by |
|---|---|---|
| inch | mm | 25.4 (exact - defined that way since 1959) |
| mile | feet | 5280 |
| mile | inches | 63,360 |
| mile | meters | 1609.34 |
| lb (force) | newton | 4.448 |
| psi | kPa | 6.895 |
| psi gauge | psi absolute | add 14.7 (sea level) |
| °C | K | add 273.15 |
| mph | ft/s | 1.4667 |
| minutes | seconds | 60 |
| rev/min | rev/s | divide by 60 |
Worksheet
Show units on every line. An answer with no unit is wrong.
1. The Sniper trap. The Birel baseline sheet says toe "2 mm out." The Sniper reads per side and we set it to 1 mm per side, 2 mm total. How much is 2 mm in inches? (Why it matters: the tie rods are marked in mm, the tape measure in inches, and reading one as the other flips toe-out into toe-in.)
2. Where the 60 and the 63,360 come from. Speed in mph = (engine rpm / gear ratio) x (tire circumference in inches) x 60 / 63,360. Write the formula out as four steps, one multiplication or division per line, with the unit of every number (rev/min, in/rev, min/hr, in/mi). Cross off each unit that appears on top of one line and underneath the next. What is left at the end should be mi/hr. Then say in one line what the 60 converts and what the 63,360 converts.
3. Top speed on the 72. Tire OD 10.9 in (circumference = pi x 10.9 = 34.24 in). Gearing 72/20 = 3.600. Limiter 6100 rpm. What is the kart's speed on the limiter, in mph? Then the same for the 70-tooth rear (ratio 3.500). How many mph faster is the 70 when both are on the limiter? (Check yourself against our speed calculator. Both "10.9 in" and "6100 rpm" are unverified inputs - the MyChron will fix the second one.)
4. Average vs top speed. Your best lap at Buttonwillow was 58.33 s on a 0.69 mi track. What was your average speed in mph? Your Bakersfield best was 31.14 s on 0.5 mi. What was the average speed there? Which track has the higher average speed, and by how many mph? Is that difference the track's layout or the driver? What would you need to know to tell?
5. Gauge vs absolute. The tire gauge reads pressure ABOVE atmosphere. Atmosphere is 14.7 psi. On 9/19 we set 12 psi cold and read 14 psi hot. Convert both to absolute. By what percentage did the absolute pressure rise?
6. The gauge is a thermometer. For a fixed volume of gas, absolute pressure is proportional to absolute temperature (kelvin). If the tire was 27 °C (300 K) cold, what was the air inside the tire after the session, in K and in °C? (Use your percentage from Q5. Ignore the air that leaks out on the chuck; lesson 4 Q7 is about that.) When the pyrometer arrives, check your answer against its reading of the tread and write down how far apart the two are.
7. Half a psi. The JOES gauge is rated 2.5% of full scale on a 0-30 psi dial. What is that in psi? The right-rear qualifying change was 12 -> 11.5. Is that 0.5 psi change bigger or smaller than the gauge's rated accuracy? So: can the gauge tell us for sure that the tire's pressure actually changed?
8. Four playing cards. Kart + driver + gear weighed in at (at least) 360 lb. Assume it's shared equally by four tires and the tire pressure is what holds the kart up. Contact-patch area = force / pressure. At 12 psi, what is each contact patch in square inches? At 14 psi? Everything you do in a corner goes through those four patches - do they get bigger or smaller when you add pressure?
9. Weight vs mass. The scales say "360 lb." That is a force (Earth pulling). Convert it to newtons. If the kart were on the Moon (gravity 1/6 of Earth's), what would the scales read? Would the engine find it any easier to accelerate the kart out of a hairpin? (Hint: think about which of weight and mass the engine is fighting. Lesson 3 comes back to this.)
10. Catch the error. A forum post says: "Lowering the rear sprocket from 72 to 70 raises your top speed by 2 teeth." Fix the units in that sentence so it says something true.
Done? Bring the sheet to the next track day. Q6 and Q7 get checked against the pyrometer and the gauge, and the gap between your answer and the reading is the next lesson.
Going deeper the contact patch when the tire isn't a balloon