Module 3: Kinematics · Companion · Reference · Prerequisite: K1–K16
Several tools in this module have proper names that the module never used. Here they are — because a method you cannot name is one you cannot look up.
Mahavakya
A name is how a method travels. Without one you can use a tool in the place you learned it, and nowhere else.
Why this exists. While auditing the module we found techniques being used without ever being named in the text. A student would learn the method, use it correctly, and never discover that it had a name, a history, and a place in their mathematics syllabus.
That is a real cost. It means the method cannot be searched for, cannot be recognised in another book, and cannot be asked about — and worst, it looks like a physics trick when it is a standard piece of mathematics the student already owns.
The three that went unnamed
Vieta’s formulas K8, section 5
What K8 said: that for a projectile passing the same height twice, the two times satisfy t1 + t2 = 2u/g and t1t2 = 2h/g — read straight off the quadratic without solving it.
What it is called: Vieta’s formulas, after François Viète, a sixteenth-century French mathematician. For at² + bt + c = 0:
sum of roots = −b/a product of roots = c/a
Applied to ½gt² − ut + h = 0, those give exactly the two results above.
This is on your mathematics syllabus. Knowing the name turns a physics trick into a tool you already own — and one that works on any quadratic, not only on flight times.
Separation of variables K9, section 5
What K9 did: for acceleration depending on velocity, rearranged a(v) = dv/dt into
dv / a(v) = dt
— velocity on one side, time on the other, both integrable.
What it is called: separation of variables, the standard first method for solving an ordinary differential equation. K9’s other two cases use it too: a(t) separates trivially, and v dv = a(x) dx is the same move after the chain rule has removed time.
All three of K9’s cases are one technique. The part presents them as three different moves, which is right for learning them — but they are three arrangements of a single method, and the name is what makes that visible.
The osculating circle K15, section 6
What K15 said: that R is “the radius of the circle that best fits the path at that point”.
What it is called: the osculating circle — from the Latin osculari, to kiss. It is the circle that touches the curve and matches both its slope and its curvature there, and its radius is the radius of curvature.
Its centre has a name too — the centre of curvature — and K15’s figure marks it. The whole apparatus is standard differential geometry, and knowing that opens a door: curvature, evolute and torsion all live behind it.
What the module did name
For completeness, and so the pattern is visible. These are used and named, and the names are worth holding onto.
| Technique | Where | What it does |
|---|---|---|
| Triangle inequality | K1 §6 | Why s ≥ \|Δr\|, always |
| Harmonic mean | K3 §4 | Averaging speeds over equal distances |
| Arithmetic mean | K3 §4 | Averaging speeds over equal times |
| Secant and tangent | K4 §2 | The shrinking interval that defines the derivative |
| Impulse approximation | K5 §6, K6 §7, K10 §5 | What a corner on a graph idealises away |
| Chain rule | K9 §3 | Turning a = dv/dt into v dv/dx |
| Dot product | K14 §2 | The string constraint, in one line |
| Strobe diagram | K2 §4 | Positions at equal intervals; spacing shows acceleration |
One that has no name
K6 §5 uses a check we called the degree progression: if the acceleration is a polynomial of degree n, the velocity has degree n + 1 and the position degree n + 2. A constant acceleration gives a linear velocity and a quadratic position, and so on.
It has no standard name, as far as we can find. It is a consequence of the power rule rather than a technique in its own right — which is probably why nobody bothered to name it.
It is still the fastest check in the module. Before verifying any arithmetic, verify the degrees. A cubic acceleration cannot integrate to a cubic velocity, and noticing that takes a second.
Why the names matter
A method without a name works only where you met it. You cannot search for it, cannot recognise it when a different book uses it on a different problem, and cannot ask anyone about it without describing the whole thing from scratch.
Worse, it looks like a physics trick. A student who meets the sum and product of roots in K8 and never hears “Vieta” has learned a fact about projectiles. One who hears the name has recognised a tool from their algebra course — and will reach for it next time a quadratic appears anywhere at all.
Every technique in this module is standard mathematics doing physics work. None of it was invented for kinematics. The names are how you find that out.
A note on how this was found
The first two omissions were found by accident, not by looking — which suggested there would be more, and there were. The module has since been swept for the whole class of error. If you find another, it belongs on the corrections page, and we would be glad to hear it.
Prerequisite: K1–K16 · Reading: 8 min · Examined: no — this is a reference companion