Before we cut foam or print a part, we put measured numbers into the formulas that tell us whether the change is worth it. This pack is pure substitution. Same algebra skills, real Nightmare numbers, plus a checker that flags the unit mistakes that ruin real design reviews.
Substitute measured values into multi-step formulas without losing the units.
02
Rearrange a formula to make a chosen variable the subject, then work it out.
03
Convert g to kg, cm² to m² and cm³ to m³ before substituting.
04
Check whether an answer makes sense for a 55 g car on a 20 m lane.
02From the Nightmare car
Average speed from a timing strip, mass from the scale, air density from a table, frontal area from a sketch of the body. None of it helps unless someone substitutes carefully. Our worst design-review mistake this season was a frontal area typed in cm² inside an SI formula. The drag came out ten thousand times too big, and we nearly redesigned a body that was already fine. This pack trains the habit that catches that. It uses the same symbols as LAB-01, LAB-03 and LAB-05.
Fig. 4 Substitution is not typing. It means choosing the subject, converting the units, and then asking whether the answer could be real.
03Core idea
The formula bank
v = s / t | ρ = m / V | Fd = ½ρv²CdA | Frr ≈ CrrN | KE = ½mv²
The three-line rule
Line 1: write the formula. Line 2: put the numbers in, keeping the same layout. Line 3: the answer, with a unit. Marks are lost on skipped lines, and so are design reviews, because the skipped line is where the unit mistake hides.
Rearranging
To make a different variable the subject, do the same thing to both sides until that variable is on its own. From v = s/t, multiply both sides by t to get vt = s, then divide both sides by v to get t = s/v. From Frr = CrrN, divide both sides by N to get Crr = Frr/N.
!
Unit traps in this pack: g to kg (divide by 1000), cm² to m² (divide by 10 000), cm³ to m³ (divide by 1 000 000), ms to s (divide by 1000). Convert first, then substitute.
04Three-line checker
Pick a formula, type in your measured values in whatever units you have, and the checker writes the three lines for you. It converts the units on line 2 and flags anything that looks wrong. Change the subject to practise rearranging.
Units are converted to SI on line 2. A red flag means you would have been out by a power of ten.
—Result
—Rearranged form
—Sense check
05Worked example
Timing strip: s = 20.0 m, t = 1.75 s.
v = s / t
v = 20.0 / 1.75
v = 11.4 m/s (3 s.f.)
Next, use that v in the drag equation with ρ = 1.20, Cd = 0.28 and A = 45 cm², which is 0.0045 m².
Sense check: 11.4 m/s is 41 km/h, which is believable for a slow run. 0.098 N is a tenth of a newton, which is believable for a model car. If A had stayed in cm², Fd would come out as 983 N, the weight of a motorbike. An answer that is wrong by a power of ten means a unit slipped.
06Student questions Teacher mode is off. Answers are hidden.
Q1 · Substitute[4]
s = 18.0 m and t = 1.62 s. Find v. Then, with ρ = 1.20, Cd = 0.26 and A = 0.0050 m², find Fd.
Answer key
v = 18.0 / 1.62 = 11.1 m/s
½ρv² = 0.5 × 1.20 × 11.11² = 74.1
CdA = 0.26 × 0.0050 = 0.0013
Fd = 74.1 × 0.0013 = 0.0963 N, about 0.096 N
Q2 · Rearrange[2]
Make t the subject of v = s/t. A car averages 10.5 m/s over 20.0 m. How long did the run take?
Answer key
t = s / v
t = 20.0 / 10.5 = 1.90 s
Q3 · Units trap[3]
The mass is written down as 52 g. The density formula needs kg. Show the converted mass, then find the density if the volume is 4.0 × 10−5 m³.
Answer key
m = 52 g = 0.052 kg
ρ = m / V = 0.052 / 4.0×10⁻⁵
ρ = 1300 kg/m³, which makes sense because PLA is about 1240 kg/m³
Q4 · Sense check[2]
A student gets v = 11 400 m/s for a 20 m run. Without redoing the calculation, say what went wrong and how you know.
Answer key
The answer is 1000 times too big, so a unit slipped by a factor of 1000. Almost certainly t was typed in milliseconds (1750 ms) instead of seconds (1). 11 400 m/s is 33 times the speed of sound, so no car does that, and the size of the error tells you which unit (1).
07Printable worksheet The answer key prints as well when teacher mode is on
Team Nightmare · STEM Subject Lab
NM-LAB-04 · Substitution Practice
Name ______________________ Class __________ Date __________
Maths 30 minutes Links to LAB-01, LAB-03 and LAB-05
Formula bank
v = s/t ρ = m/V KE = ½mv²
Fd = ½ρv²CdA Frr ≈ CrrN
House rule
Line 1: the formula. Line 2: the numbers in SI units. Line 3: the answer with a unit. No calculator-only answers.
subjectsubstituterearrangeSI unitpower of ten
[2]1.s = 20.0 m and t = 1.68 s. Find v.
[3]2.Using your v from Q1, with ρ = 1.20, Cd = 0.29 and A = 46 cm², find Fd. Convert A first.
[2]3.N = 0.42 N and Crr = 0.013. Find Frr on one wheel and on four wheels.
[2]4. Rearrange.Starting from Frr ≈ CrrN, make Crr the subject. If Frr = 0.0060 N and N = 0.40 N, find Crr.
[3]5. Units.m = 58 g and v = 19 m/s. Find KE. Then write down the answer a student would get if they forgot to convert the mass, and how they should notice.
Accept any equivalent working. The marks in brackets match the student sheet.
Teacher copy Do not hand out
1. [2] v = s/t = 20.0 / 1.68 = 11.9 m/s
2. [3] A = 46 cm² = 0.0046 m²Fd = ½ × 1.20 × 11.9² × 0.29 × 0.0046 = 0.6 × 141.7 × 0.001334 = 0.113 N One mark for the conversion, one for the substitution line, one for the answer. Accept 0.11 to 0.114 N depending on how v was rounded.
3. [2] Frr = 0.013 × 0.42 = 0.00546 N (one wheel). 4 × 0.00546 = 0.0218 N.
4. [2] Crr = Frr / N = 0.0060 / 0.40 = 0.015
5. [3] m = 0.058 kg. KE = ½ × 0.058 × 19² = 10.5 J Forgetting the conversion gives 10 469 J, which is 1000 times too big (1). They should notice because a 58 g model car cannot carry 10 kJ, and a factor of exactly 1000 points straight at g to kg (1).