Module 3: Kinematics · Companion · Beyond the syllabus · Prerequisites: K4, K15
The chain does not stop at acceleration. Nothing in the mathematics says it should — so why does physics stop there?
Mahavakya
A derivative is not inherently velocity. It becomes velocity because position is what we chose to differentiate.
This is a companion piece, not a part. Nothing here is examined, and K1–K16 stand complete without it. It is here because the question it answers — why stop at the second derivative? — is the obvious one to ask after K4, and no textbook answers it.
On this page
1. The obvious question
K4 built a chain. Differentiate position and you get velocity; differentiate velocity and you get acceleration. Then the module stopped, and for sixteen parts never went further.
But the mathematics does not stop. If x(t) is smooth, it has a third derivative, a fourth, and as many as you like. Nothing in calculus marks the second one as special.
x → v → a → ? → ? → …
So the stopping point is a decision, not a discovery. This piece is about what lies past it, and why the decision is nonetheless a good one.
2. Jerk, and where you have felt it
The third derivative has a name and a use.
jerk j = da/dt = d³x/dt³ units m/s³
You notice acceleration, but you complain about jerk. A lift that accelerates smoothly to speed feels fine; one that starts abruptly is unpleasant, and the difference between them is entirely in how quickly the acceleration itself changes.
This is why railway curves are not circular arcs. A straight track has zero centripetal acceleration; a circular curve has v²/R. Joining them directly makes the acceleration jump discontinuously at the join — infinite jerk, and a lurch you can feel.
So real track uses a transition spiral, whose curvature grows linearly along its length. The acceleration then rises linearly too, and the jerk is constant and finite. Roller coasters are designed the same way, and for the same reason.
Look back at K5’s corners with this in hand. A corner on a position–time graph is a jump in velocity — infinite acceleration. A corner on a velocity–time graph is a jump in acceleration — infinite jerk. The module drew plenty of those, and called them idealisations. This is what they were idealising away.
3. Snap, crackle and pop
The names past jerk are real, and cheerfully unserious.
| Order | Name | Units | Used? |
|---|---|---|---|
| 1st | velocity | m/s | everywhere |
| 2nd | acceleration | m/s² | everywhere |
| 3rd | jerk | m/s³ | engineering — track and ride design |
| 4th | snap (jounce) | m/s⁴ | rarely |
| 5th | crackle | m/s⁵ | essentially never |
| 6th | pop | m/s⁶ | essentially never |
The last three are named after a breakfast cereal’s mascots. Physicists are not always solemn.
4. Why physics stops at two
Not because the higher derivatives fail to exist. Because of where the physics enters.
F = ma
Newton’s law relates force to the second derivative, and no higher one. So a force law plus two initial conditions — a position and a velocity — determines the entire future of a particle. Everything beyond acceleration is already implied by that, not independently specified.
Two initial conditions, and exactly two. Give a particle a starting position and a starting velocity and its trajectory is fixed. You do not get to choose the initial jerk — the force law has already decided it.
That is why K9 needed a constant of integration at each of two steps and no more, and why K8’s four equations take u and x₀ as their only inputs. The number two in “second-order” and the number two in “two constants of integration” are the same two.
So the chain stops where it does because of a fact about nature, not about mathematics. Had the fundamental law been third-order, jerk would be as ordinary a word as acceleration, and this would not be a companion piece.
5. The chain in oscillation
Simple harmonic motion shows the whole chain at once, because differentiating it returns you to where you began.
Take x = A sin ωt and differentiate repeatedly:
x = A sin ωt
v = Aω cos ωt
a = −Aω² sin ωt = −ω²x
j = −Aω³ cos ωt = −ω²v
s = Aω⁴ sin ωt = −ω²a
Every differentiation multiplies by −ω² and shifts the phase by a quarter cycle. So the chain closes on itself with period four: x, a and s share one shape, v and j share the other.
Watch the signs — every step carries a minus. It is easy to write s = +ω²a by pattern-matching against x and s both having a sine, and forgetting that four factors of −ω² do not cancel against two. Check numerically if in doubt: with A = 2, ω = 3 and t = 0.4, both s and −ω²a come to +151.0, while +ω²a gives −151.0.
K15 met the first of these relations already: a = −ω²r for uniform circular motion. That was the same identity, arrived at from geometry rather than from repeated differentiation.
6. What the chain is actually about
There is nothing in the operation d/dt that knows it is producing a velocity. It takes a function of time and returns its rate of change. What comes out is a velocity only because what went in was a position.
Differentiate a temperature and you get a heating rate. Differentiate a charge and you get a current. Differentiate a population and you get a birth rate net of deaths. The operation is identical; only the meaning of the input differs — and the meaning of the output follows from it.
So the names in this chain — velocity, acceleration, jerk — are labels for positions in a sequence, not for different mathematical objects. There is one operation here, applied six times.
Which is worth carrying into Module 4. Force will be introduced as something that causes acceleration, and it is easy to start believing the equation does the pushing. It does not. The equation records a regularity someone noticed. The pushing is done by the world.
Worth keeping
- The chain does not end at acceleration. Physics stops there because F = ma is second-order, not because the derivatives run out.
- Two initial conditions, because the law is second-order. The same two that appear as constants of integration in K9.
- Jerk is real engineering: transition spirals on railways and roller coasters exist to keep it finite.
- An infinite jerk is what a corner on a velocity–time graph idealises away, just as infinite acceleration is what a corner on a position–time graph idealises away.
- In SHM every derivative multiplies by −ω². Each step carries a minus sign, including s = −ω²a.
- A derivative has no intrinsic meaning. It inherits one from what was differentiated.
Prerequisites: K4, K15 · Reading: 10 min · Examined: no — this is a companion piece