Mechanics 1

Deep Dive: Kinematics Constant Acceleration Two Stage Motion

Provenance: written 2026-09-29. Read in full for this pass: research/veridian/WME01-verified-facts.md (§1–§9, grepped specifically for "kinematics"/"two-stage"/"suvat" and cross-read around each hit); content/veridian/curriculum/wme01/kinematics-constant-acceleration-two-stage-motion.ts (the lesson this document is written not to duplicate — its diagram, marked-solution, worked-chain and all six MCQs were read in full before anything below was drafted); content/veridian/technique-hub/wme01/codex-formula-and-method-bank.md §1.4/§2.4–2.6, forge-exam-technique-guide.md (in full), and signal-frequency-and-boundaries.md (in full); content/veridian/question-bank/wme01.ts (checked for overlap — its one two-stage item, wme01-jan25-q7c-two-stage-pulley-kinematics, is a connected-particles/4.2 item, not a duplicate of this paper's own spec-3.1 scope). No new primary-source PDF was fetched this pass — every exam-specific claim below traces to one of the files just named, each already checked line-for-line against the real Pearson PDFs in a prior pass. Every number in every micro-example below is VERIDIAN-original and was checked by exact-fraction/symbolic arithmetic (Python fractions/sympy) before being written in.

#CODEX Deep-Dive: Kinematics — Constant-Acceleration Two-Stage Motion (WME01, spec 3.1)

#Why this document exists

The lesson file already teaches this topic well: it derives the five suvat equations from one velocity–time graph, drills the dominant single-suvat-equation trap through a worked chain, a 9-mark marked solution, and six MCQs, and closes with a reference card. This document does not re-teach any of that. Its job is the layer above and below the lesson — the fuller theoretical picture (a second, calculus-based derivation route the lesson doesn't use; the formal continuity conditions that make "two-stage" well-defined at all; the three graph representations the spec's own guidance text names but the lesson only builds one of) and a complete, honestly-scoped synthesis of every real exam angle this specific topic has actually been tested from, with each claim traced to a named series rather than asserted in general terms.

Where a sub-case is already fully illustrated in the lesson, this document states the general principle once and points there rather than re-running the same scenario with cosmetic changes. Where a sub-case has no worked illustration anywhere in this repo yet, a short (2–6 line) VERIDIAN-original micro-example is given — deliberately shorter than a real past-paper question, since the job here is to isolate the mechanism, not to drill exam timing.

#1. Scope: what "two-stage motion" actually is, and what it isn't

Spec item 3.1 covers "motion in a straight line with constant acceleration" only (WME01-verified-facts.md §1). Two boundaries follow directly from that wording, both worth stating explicitly because each one is a real, confirmed trap in its own right:

A change of acceleration value is what forces a second equation — not a reversal of velocity's sign. A particle thrown straight up under gravity alone decelerates, stops, then falls back down, all under one constant acceleration ( $-g$ ) the entire time; one suvat equation (with signed $u$ , $s$ , $v$ ) describes the whole flight. Nothing about this is two-stage. The lesson's own prequestion 2 already isolates this distinction (choice C: "the direction of travel is not what forces the split... a change in acceleration's sign without any reversal of motion would force exactly the same split") — this deep-dive states it as the topic's own defining boundary, not just a distractor to avoid.
Two-dimensional motion under a constant velocity plus a separately-accelerating component (classical "projectile motion") is not this spec's content at all. codex-formula-and-method-bank.md §1.3 already establishes this as a genuine correction to an assumption a prior task brief made: spec 3.1 is restricted to a straight line, and vector kinematics on this paper (spec 2.2) uses constant-velocity position vectors $\mathbf{r}(t) = \mathbf{r}_0 + t\mathbf{v}$ , never a 2-D suvat combination with gravity. Two-stage motion, as WME01 tests it, is always a straight-line problem: one axis, one changing scalar acceleration.

What does force a genuine two-stage split, confirmed across the archive: a force added or removed partway through the motion (a parachute opening — Jun 2024 Q5); a change in the physical cause of the acceleration (a vehicle moving through distinct phases — Jan 2023 Q5; a surface change, ramp to floor, in this course's own worked examples); or the motion itself hitting a stated boundary before its current stage would naturally end. None of these require the object to change direction — the lesson's central trap is exactly the failure to notice one of these triggers has occurred and instead running one suvat equation across the whole thing.

#2. The five equations, derived two ways

The lesson derives all five suvat equations from one velocity–time graph (gradient = acceleration, area = displacement) — a genuinely complete derivation, and the one worth having first, since it needs no calculus. There is a second, equally short derivation route the lesson doesn't use, and it's worth having as well, because it is the one that generalises: the differential-equation view.

Acceleration is defined as the rate of change of velocity, and velocity as the rate of change of displacement: $a = \frac{dv}{dt}$ , $v = \frac{ds}{dt}$ . Holding $a$ constant and integrating once, starting from $v = u$ at $t=0$ :

v(t) = u + \int_0^t a \, d\tau = u + at

— the first suvat equation, this time read off an integral rather than a gradient. Integrating a second time, starting from $s=0$ at $t=0$ :

s(t) = \int_0^t v(\tau)\, d\tau = \int_0^t (u + a\tau)\, d\tau = ut + \tfrac{1}{2}at^2

— the second equation. The remaining three follow exactly as the lesson's own mechanism block already shows (eliminate a variable between these two), so there is no need to re-derive them a second way here; the point of doing it via integration rather than area is that this route is the one that generalises directly to a two-stage problem, because it makes explicit what a "stage" mathematically is: an interval on which $a$ is a genuine constant, over which the integral is valid. The moment $a$ stops being constant — even if it's still piecewise constant, just at a new value — the single antiderivative $ut + \tfrac12 at^2$ stops being a truthful description of $s(t)$ beyond that point. This is the same fact the lesson's own header states in prose ("a suvat equation spanning both phases is describing a journey that never happened"); the calculus form is the same fact stated as a domain restriction on an integral, which is what it actually is.

#3. What "two-stage" means formally: continuity of $s$ and $v$ , discontinuity of $a$

A two-stage journey is exactly this: a piecewise-constant acceleration function,

a(t) = \begin{cases} a_1 & 0 \le t < t_1 \\ a_2 & t_1 \le t \le T \end{cases}, \qquad a_1 \ne a_2,

together with the physical requirement that a real particle's position and velocity cannot jump — only its acceleration can. Formally: $s(t)$ and $v(t)$ are both continuous at $t=t_1$ (the particle doesn't teleport, and its speed doesn't instantaneously change), but $a(t)$ itself is discontinuous there by definition — that discontinuity is the whole content of "stage 2 has started."

This is exactly the handover mechanism the lesson already teaches operationally — one stage's own output value becomes the next stage's own input value — stated here as the formal reason it's allowed: continuity of $v$ at the join is not an assumption being smuggled in, it is the physical content of "no instantaneous change of speed," which is true of every scenario this topic covers (a force changing, a surface changing, or a stage ending at a boundary — none of these are collisions, which is a different physical model entirely and sits under spec 4.2(ii), not 3.1). Two direct consequences worth stating explicitly, because both are genuinely tested (§5 below):

Stage 2's own $u$ is stage 1's own final $v$ — not $0$ , and not any other value — because $v$ is continuous at the join.
Stage 2's model is only a truthful description of the motion for as long as stage 2's own physical cause is still acting. There is nothing in the mathematics of a suvat equation that knows when to stop being true; that has to come from re-reading the stem for where stage 2's own boundary actually is (§5.2 and §5.5 below).

#4. Three graph representations of the same journey — and why only one of them shows a "corner"

Spec item 3.1's own guidance text names four graph types together: "displacement-time, velocity-time, speed-time and acceleration-time graphs" (already quoted in the lesson's teach block, from WME01-verified-facts.md §1). The lesson builds one of the four (velocity–time, matching its own worked scenario) in full diagram form. The other three are real, spec-named content worth having explicitly, because each one shows the join between stages differently, and knowing which graph shows what stops a candidate reading one graph's shape onto another.

Velocity–time graph: a straight segment of gradient $a_1$ , then a second straight segment of gradient $a_2 \ne a_1$ , meeting at a corner — a genuine kink, not a smooth bend, because the gradient itself (which is $a(t)$ ) jumps at $t_1$ . This is the representation the lesson uses, and its diagram's own commonError block already establishes that the two segments have no reason to share a gradient.
Displacement–time graph: $s(t)$ is a quadratic (a parabola) on each stage, and at $t_1$ the two parabolic arcs join with a continuously turning tangent — no corner — because $s(t)$ 's gradient is $v(t)$ , which is continuous at the join by §3 above. What does change abruptly at $t_1$ is the arc's curvature (its second derivative, which is $a(t)$ ), not its slope. A candidate expecting the same visible kink they'd draw on a v–t graph, drawn instead on an s–t graph, is drawing a feature the mathematics doesn't actually produce.
Acceleration–time graph: the starkest of the three, and the most literal picture of what "two-stage" means — a horizontal segment at height $a_1$ , a vertical jump to height $a_2$ at $t_1$ , then a second horizontal segment. There is nothing to solve on this graph; it is the one representation where the defining feature of two-stage motion (a genuinely discontinuous acceleration) is drawn directly rather than inferred from a gradient or a curvature change.
Speed–time vs. velocity–time: speed is $|v(t)|$ , so the two graphs are identical for any journey where $v(t)$ never crosses zero — true of every confirmed two-stage scenario in the archive (accelerate, then decelerate, either to rest or to a stated boundary; the particle never reverses direction mid-journey in any real question this facts bank has verified for this topic). The two representations would diverge only if a stage's deceleration continued past $v=0$ into reversed motion, at which point the velocity–time graph would run through the time axis into negative values while the speed–time graph reflects back up (a "V" shape) — a real, well-defined possibility of the underlying mathematics, but not one this course's reviewed archive has confirmed as an actual WME01 question for this topic, and it is not presented as one here.

#5. Confirmed exam sub-cases, each with a short micro-example

Five distinct real sub-cases are confirmed in the reviewed archive for this specific spec point (a sixth, closely related sub-case that combines this topic with connected-particles dynamics, is scoped to a sibling lesson — see §5.6). Each entry below states exactly which series it is confirmed in; none is described as more frequent than the named citations actually support.

#5.1 The dominant trap: a two-phase journey run as one suvat equation

Confirmed in two series — Jan 2023 Q5 (a vehicle "in two distinct phases of acceleration," where candidates "tried to use $t=14$ in a single suvat equation" and were told this achieved no credit) and Jun 2024 Q5 (a box under a parachute, where some candidates "failed to realise there were two distinct stages of the motion"). This is this topic's own spine, already given full treatment in the lesson's worked-chain, marked-solution and lead trap-taxonomy item — no further worked example is added here.

The general mechanism, stated once: whenever the stem names or implies two distinct physical causes acting in sequence (not simultaneously), resolve the first cause's stage completely on its own three known quantities, extract the single handover value, then resolve the second stage as an entirely fresh suvat problem with its own $u$ (the handover value), its own $a$ , and whatever else the stem gives for stage 2 alone.

Micro-check (VERIDIAN-original, distinct from every scenario in the lesson): a cyclist accelerates uniformly from $4$ m s $^{-1}$ to $10$ m s $^{-1}$ over $3$ s, then immediately decelerates uniformly to rest, covering a further $25$ m. Find the total distance and total time. Stage 1: $a_1 = \frac{10-4}{3} = 2$ m s $^{-2}$ ; $s_1 = \frac{1}{2}(4+10)(3) = 21$ m. Stage 2: $u=10$ (the handover — not $0$ ), $v=0$ , $s_2=25$ m, so $0 = 10^2 - 2a_2(25) \Rightarrow a_2 = -2$ m s $^{-2}$ ; $t_2 = \frac{0-10}{-2} = 5$ s. Total distance $= 21+25 = 46$ m; total time $= 3+5 = 8$ s. (Both stages happening to share the magnitude $2$ m s $^{-2}$ is a coincidence of these chosen numbers, not a rule — the two accelerations have no reason to match in general, exactly as the lesson's own diagram commonError block already establishes for its own numbers.)

#5.2 A second-stage variant: the stage ends at a stated boundary, not at rest

Confirmed in two series, independently cross-checked in the lesson's own audit trail: Jun 2024 Q5 part (b) scores the A-mark for a landing speed of $v=4$ m s $^{-1}$ , not $0$ , because the box reaches the ground — a positional boundary — before its parachute-deceleration would otherwise bring it to rest; part (d)'s own marking notes penalise a graph that "ends on the t-axis" or assumes "it stops as it reaches the ground." Jan 2023 Q5 part (c) independently corroborates the same general shape with a time boundary instead of a positional one (asking for the distance at a stated $t=14$ , not at the moment the vehicle would naturally stop). This sub-case is already fully illustrated in the lesson's MCQ 6 (crate and loading bay); the general principle is worth stating on its own terms because it is easy to state and easy to forget under pressure: nothing in a suvat equation knows where the stem's own boundary is — that information exists only in the words of the question, and has to be checked against the stage's own natural stopping point (found separately, by setting $v=0$ ) before assuming the two coincide.

Micro-check (a different domain from the lesson's own crate scenario, to show the same mechanism doesn't always land on a clean number): a skier accelerates uniformly from rest at $1.6$ m s $^{-2}$ for $5$ s down a slope, then crosses a flat run-out zone $25$ m long, where friction decelerates them at $1$ m s $^{-2}$ , before a safety barrier. Find their speed at the barrier. Stage 1: $v = 1.6(5) = 8$ m s $^{-1}$ ; $s_1 = \frac{1}{2}(1.6)(25) = 20$ m. Stage 2: the skier's true stopping distance, if the run-out zone were long enough, would be $\frac{8^2}{2(1)} = 32$ m — more than the $25$ m actually available, so the barrier is reached first, still moving: $v^2 = 8^2 - 2(1)(25) = 64-50 = 14 \Rightarrow v = \sqrt{14} \approx 3.74$ m s $^{-1}$ . The answer is deliberately not a clean number here — a real two-stage boundary-cutoff question does not owe you one, and a surd or a rounded 3 s.f. value is a perfectly legitimate final answer, not a sign of an error.

#5.3 A three-phase graph with an unknown time, solved by area

Confirmed — Jan 2023 Q1 (accelerate, travel at constant speed, decelerate, over a stated total distance with one phase's duration unknown, $T$ ); the graph itself carries its own B-marks for correct shape, and a documented, specific arithmetic trap ("halve the time taken to decelerate rather than double it"). This sub-case already has a full worked short-check in codex-formula-and-method-bank.md §2.6 (a particle accelerating to 20 m/s in $T$ s, cruising 15 s, decelerating over 4 s, total 500 m, solving $T=16$ s) and is not repeated here. Worth naming in this synthesis for completeness, and because it is structurally a three-phase version of the same handover discipline: the middle (constant-velocity) phase has zero acceleration, contributes a simple rectangle to the total area, and needs no suvat equation of its own at all — only $s = vt$ .

#5.4 Two particles on offset clocks

Confirmed — Jan 2020 Q3 (a particle thrown up, a second particle released later, asked when/where they meet; candidates "wrote down two correct expressions for the displacements" but, per the same report, mismatched the two particles' own starting times when comparing them). This is a related but distinct variant from §5.1–5.3: it is not one particle changing acceleration, but two particles under the same constant acceleration model, whose clocks don't share an origin — the handover discipline generalises from "carry $v$ across a stage boundary" to "shift $\tau = t - \Delta t$ before comparing two displacement expressions built on different clocks." Already fully illustrated with a worked scenario in the lesson's MCQ 3; the general substitution rule is stated here once rather than re-illustrated: if particle $Y$ starts $\Delta t$ after particle $X$ , $Y$ 's own displacement expression must be evaluated at $\tau = t-\Delta t$ , not at $t$ itself, before it can be validly set equal to $X$ 's.

#5.5 A quadratic root outside the model's own valid domain

Confirmed — Oct 2019 Q6 (solving a suvat-derived quadratic left two roots, only one physically valid; candidates "failed to explain clearly why they were rejecting" the invalid one, and separately were reminded to show working when solving quadratics). This connects directly to §3's own formal statement: a stage's $s(t)$ formula is a truthful description of the motion only within that stage's own domain — typically $0 \le t \le$ (the time at which $v=0$ , or the time the stage's own boundary is reached). A root of the equation $s(t) =$ (some target value) that lies outside that domain is not describing a physically real moment; it is describing what the same parabola would say if extended beyond where it stopped being a model of anything. Already fully illustrated with a worked scenario in the lesson's MCQ 4 (roots at $t=2$ and $t=8$ , domain ending at $t=5$ ); the domain-validity principle is stated here in its general form, with one compact independent check:

Micro-check: a stage's displacement is modelled by $s(t) = 6t - 0.5t^2$ , valid while the particle is still moving under this deceleration — i.e. until $v(t) = 6-t = 0$ , at $t=6$ s, so the model's own domain is $0 \le t \le 6$ . Solving $6t-0.5t^2=16$ gives $t^2-12t+32=0 \Rightarrow t=4$ or $t=8$ . Since $8$ lies outside the domain $[0,6]$ , only $t=4$ s is a valid answer — not because $8$ is "too large" as a matter of instinct, but because the function $s(t)=6t-0.5t^2$ stops being a description of this particle's motion the moment $t$ passes $6$ .

Confirmed in two series — Jan 2025 Q7 (a connected-particles pulley system runs for a stated distance, then the lighter particle continues "moving under gravity" rather than at the pulley system's own shared acceleration, for a further stated distance; candidates who used "only one equation for the whole of the motion" made no progress) and Jun 2025 Q8 (per signal-frequency-and-boundaries.md's topic-frequency table, an equivalent structural combination, not itself deep-mined for qualitative content in this pass — flagged honestly rather than assumed identical to Jan 2025's). forge-exam-technique-guide.md names this exact shape — connected particles, then a free-motion continuation — as consistently the paper's last and heaviest question in both fully mark-mapped recent series (15 of 75 marks in Jan 2025, 17 of 75 in Jun 2025), while itself flagging this as a two-series pattern, short of the 3+-series bar this course otherwise wants before calling something a standing convention.

This sub-case genuinely tests the same handover discipline this whole document is about — one stage's output ( $v$ at the moment the string goes slack or a particle lands) becomes the next stage's input, and the acceleration changes value (from the shared pulley acceleration to plain $g$ ) exactly the way §3 formalises. It is not developed further here because it is squarely spec 4.2's own territory (a connected-particles system, not this document's spec-3.1 remit) and already has a full 15-mark worked solution, its own trap-taxonomy item, and an MCQ in connected-particles-pulleys-and-lifts.ts — named here only so the synthesis in §6 doesn't silently omit it.

#6. Synthesis: every real exam angle this topic has been tested from

Consolidating §5 into one list, each entry naming exactly what is confirmed and in how many series — no likelihood is stated as a percentage anywhere below, only counts of series actually checked, consistent with this course's standing rule.

Single-suvat-equation-across-two-stages — confirmed Jan 2023 Q5, Jun 2024 Q5 (two series; a related but distinct pattern also appears once each in Oct 2019 and Jan 2020, not counted here as the same trap per signal-frequency-and-boundaries.md's own precise accounting).
Second stage cut short by a stated boundary rather than run to rest — confirmed Jun 2024 Q5 (positional/ground boundary) and Jan 2023 Q5 part (c) (time boundary) — two series, two different boundary types.
Graph-shape errors on a multi-phase velocity–time graph with an unknown time (symmetric segments drawn where the real shapes differ; the deceleration-time ratio inverted) — confirmed Jan 2023 Q1.
Offset start-times between two particles under the same clock — confirmed Jan 2020 Q3.
A quadratic root outside the model's own valid time domain, rejected without stated justification — confirmed Oct 2019 Q6.
Connected-particles-then-free-motion, the same handover discipline combined with spec 4.2 — confirmed Jan 2025 Q7, Jun 2025 Q8 (two series; owned by a sibling lesson, see §5.6).
Topic frequency, from the two fully mark-mapped recent series: spec 3.1 anchors real marks in both Jan 2025 (Q2, 11 marks; Q3, 12 marks, shared with 4.3–4.4; Q7, 15 marks, shared with 4.2) and Jun 2025 (Q3, 8 marks; Q7, 13 marks, shared with 2.1–2.2; Q8, 17 marks, shared with 4.2) — per signal-frequency-and-boundaries.md §1. Flagged honestly: this confirms spec 3.1 anchors marks in both series; it does not confirm what error pattern Jan 2025 Q2/Q3 or Jun 2025 Q3/Q7 individually test, since neither series' examiner report has been deep-mined for this topic's own qualitative content in any pass this repository records. No trap is invented to fill that gap.
A general sign-convention trap independently reproduced on a kinematics question (not specific to two-stage motion, but recurring on this spec item): Jan 2025's report records that on a kinematics deceleration question, "a substantial number gave a negative answer and lost a mark" (per forge-exam-technique-guide.md) — the same discipline the dedicated momentum-impulse-sign-convention.ts lesson covers in depth for spec 4.3, now independently confirmed to recur here too.
A found value not carried forward into a later kinematics part — Jan 2025 Q7(b), after a connected-particles acceleration was correctly found: "a common error (scoring no marks) was to use g for the acceleration" instead of the smaller, already-derived system value (per forge-exam-technique-guide.md). This is the same handover discipline as §3 above, misapplied one part later than usual — the "carried value" being lost is not a stage-to-stage velocity but a part-to-part acceleration.

#7. What remains genuinely thin, stated honestly

Jan 2025 Q2/Q3 and Jun 2025 Q3/Q7's own qualitative content for this specific spec point is not yet available anywhere in this repository — signal-frequency-and-boundaries.md confirms these questions exist and carry real marks against spec 3.1, but was explicitly scoped to frequency and grade boundaries, not per-question error analysis, for these two series. This document does not invent what those questions' own traps might be.
No genuine WME01 example of a two-stage journey where the particle reverses direction mid-second-stage (the case flagged in §4's speed-vs-velocity-graph discussion) has been found in the 17-series archive WME01-verified-facts.md covers. The theoretical point stands (it follows from the definition of speed as $|v|$ ), but no real question is cited as testing it, and none is invented here.
A full per-series count beyond "two" for the dominant §5.1 trap was not attempted here, matching the lesson's own header correction of an earlier "three series" overclaim in this same facts bank — this document inherits that correction rather than restating the higher, unsupported number.

#Sources

research/veridian/WME01-verified-facts.md (§1, §3, §4 topic 3, §6, §9's own audit entry for this lesson file); content/veridian/curriculum/wme01/kinematics-constant-acceleration-two-stage-motion.ts; content/veridian/technique-hub/wme01/codex-formula-and-method-bank.md (§1.4, §2.4–2.6); content/veridian/technique-hub/wme01/forge-exam-technique-guide.md; content/veridian/technique-hub/wme01/signal-frequency-and-boundaries.md; content/veridian/question-bank/wme01.ts (checked for overlap only). Not affiliated with or endorsed by Pearson Edexcel; every direct quotation above is a fragment under 15 words with its own series/question attribution, and everything longer than that has been paraphrased rather than reproduced.