Module 3: Kinematics · Reference · Prerequisite: K1–K15
Fifteen parts, one structure. This page is for looking things up and for checking that you know which condition each formula depends on.
Mahavakya
Calculate only what is required. Do not calculate simply because the mathematics allows you to.
On this page
1. The one structure
Strip away the topics and kinematics is a single chain, walked in either direction.
Going right needs nothing. Going left needs an initial condition at every step.
Everything in the module is this chain under a different condition, in a different coordinate, or for a different number of bodies:
| Description | What it looks like | Parts |
|---|---|---|
| One dimension | signed scalars | K3–K10 |
| Two dimensions | component by component | K11, K12 |
| Graphical | slopes down, areas up | K5, K6 |
| Relative | the chain applied to differences | K7, K13 |
| Angular | θ instead of x | K15 |
These are not five theories. They are five projections of one structure, which is why a technique learned in one reappears in another.
2. Which tool does this problem need?
Twelve physical questions and the mathematics each one calls for. Reading a problem is largely the business of deciding which row you are in.
| The question asks… | Use | Part |
|---|---|---|
| Where is it? | evaluate x(t) | K1 |
| How fast, over an interval? | Δx/Δt or s/Δt | K3 |
| How fast, right now? | dx/dt | K4 |
| When is it at rest? | solve v = 0 | K4 |
| Does it turn round? | v = 0 and sign change | K4 |
| Speeding up or slowing? | sign of va | K4 |
| Total distance travelled? | split at turning points, sum \|Δx\| | K6 |
| Displacement from a graph? | signed area under v–t | K6 |
| Where does it stop, no time given? | v dv = a dx | K9 |
| Two bodies meeting? | xA(t) = xB(t), same t | K7, K13 |
| Bodies linked by a string? | differentiate the length constraint | K14 |
| Curved path, constant speed? | an = v²/R, at = 0 | K15 |
3. The formulas, with their conditions
Every formula below is followed by the condition it depends on. A formula without its condition is not knowledge; it is a liability.
Definitions — always true
| v = dx/dt | definition of velocity | K4 |
| a = dv/dt = d²x/dt² | definition of acceleration | K4 |
| a = v dv/dx | chain rule; eliminates t | K9 |
| s ≥ \|Δr\| | equality only if no reversal | K1 |
Constant acceleration only
| v = u + at | K8 |
| s = ut + ½at² | K8 |
| v² = u² + 2as | K6, K8 |
| s = ½(u + v)t | K8 |
These four fail the instant the acceleration varies — and no average value rescues them. The general form is v² = u² + 2∫a dx.
Projectiles — equal launch and landing height only
| T = 2u sin θ / g | K12 |
| R = u² sin 2θ / g | K12 |
| H = u² sin² θ / (2g) | this one holds generally |
| y = x tan θ − gx²/(2u²cos² θ) | trajectory, holds generally |
For a cliff, a slope, or any unequal landing height: set y to the actual landing height and solve the quadratic.
Vectors, relative motion, constraints, circles
| vBA = vB − vA | always; B observed by A | K7, K13 |
| aA = aB ⇒ vBA constant | relative motion uniform | K13 |
| vA·n̂ = vB·n̂ | taut inextensible string | K14 |
| v = Rω | fixed radius | K15 |
| an = v²/R, at = dv/dt | any curved path; R is local | K15 |
| \|a\| = √(at² + an²) | perpendicular components | K15 |
4. Every trap in one place
- Negative acceleration means slowing down. Only if v > 0. Use the sign of va. K4
- v = 0 means a turning point. Necessary, not sufficient. Check the sign either side. K4
- a = 0 means at rest. It means the velocity is not changing. K4
- Averaging the two speeds. Right only for equal times; equal distances give the harmonic mean. K3
- Ignoring rest periods. They add to the total time and nothing to the distance. K3
- Height on a graph read as slope. Height is the quantity; slope is its rate. K5, K6
- Area under a–t taken as velocity. It is the change in velocity. K6
- Forgetting to reject a negative time. A correct equation can give an inadmissible answer. K7
- Acceleration zero at the top of a throw. It is −g throughout. K8
- Release from a moving platform taken as from rest. It inherits the platform’s velocity. K8
- Constant-acceleration equations with varying a. Go back to the definitions. K9
- Same limits on both sides of v dv = a dx. Velocities left, positions right. K9
- Resetting the clock or the velocity at a stage boundary. Neither resets. K10
- Taking the magnitude before differentiating. Components first, magnitude last. K11
- The ball stopping at the apex. Only vy = 0; vx is untouched. K12
- Range formula on an unequal-height launch. Set y to the real landing height. K12
- Umbrella tilted away from the walking direction. Tilt toward. K13
- Assuming a movable pulley gives 2. Count the segments. K14
- Treating the string angle as constant. It changes as the bodies move. K14
- Constant speed taken as zero acceleration. Only if the direction is constant too. K15
- Adding at and an arithmetically. They are perpendicular. K15
5. What Module 1 supplied
Worth seeing in one table, because the answer to “why did we do all that mathematics first” is here.
| Foundation | Where it did the work |
|---|---|
| 1 — modelling | every idealisation; the corner that hides an impulse (K5) |
| 2 — scalars and vectors | speed against velocity (K3); the whole of K11 |
| 3 — coordinates | frames and origins (K1); axis choice (K12) |
| 4 — vector algebra | the triangle inequality (K1); component differentiation (K11) |
| 5 — dot product | the string constraint (K14) |
| 6 — cross product, axial vectors | angular velocity along the axis (K15) |
| 7 — graphs | slope and area (K5, K6); y(x) against y(t) (K12) |
| 8 — differentiation | the limit that defines velocity (K4); rotating basis (K15) |
| 9 — integration | areas as accumulation (K6); all of K8 and K9 |
| 10 — approximation | small-angle step in the centripetal derivation (K15) |
| 11 — the toolkit | v dv = a dx, named there as worth memorising (K9) |
6. The method
- Read once without writing. Decide what kind of motion this is.
- Draw. Origin, positive direction, every known quantity, the unknown marked. K2
- Ask what the acceleration depends on. Nothing, t, x, or v — this chooses your method. K9
- Ask what the question actually wants. A position, a time, a distance, a condition.
- Look for structure before calculating. Do both bodies share an acceleration? Would another frame make this a one-body problem? Is there a symmetry?
- Now write equations — and only those the question needs.
- Check admissibility. Is t positive? Is the answer on the part of the path that exists? Does it agree with the drawing?
Steps 3 and 5 are what separate a fast solution from a long one. Step 5 in particular: the collision problem in K13 collapses to one line once you notice both particles share g, and stays a page of algebra if you do not.
7. Practising in the right order
Working harder problems before the easier kinds are secure wastes both. The levels are not difficulty bands but different skills.
- Recognition — which case is this? No calculation.
- Direct application — the condition is stated, the formula fits.
- Reconstruction — derive the relation you need rather than recalling it.
- Constraint — the problem gives a condition, you translate it into an equation.
- Structure — two bodies, a frame choice, a symmetry to exploit.
- Modelling — decide what to idealise before any physics starts.
Contemplation
Look back at what the module did. A moving object became a function. A constraint became an equation. A collision became the equality of two positions at one time. A graph became a story about a journey.
None of those translations was forced on us by the world. We chose them, and we chose them because they work — because the mathematics turns out to have the same shape as the thing being described, closely enough that manipulating the symbols predicts what the object will do.
Foundation 1 said mathematics does not govern the world but records the regularities we find in it. Fifteen parts later that is still the right description — and it is worth holding on to, because the next module adds force, and it is very easy to start believing that the equations are doing the pushing.
What comes next
Kinematics describes motion without asking what causes it. Every acceleration in these sixteen parts was given, assumed, or read off a graph — never explained.
Module 4 asks where accelerations come from. The description you now have is what makes that question answerable: Newton’s second law relates force to acceleration, and acceleration is a thing you can now compute, measure and reason about in any coordinate system, for any path, for linked or free bodies.
Prerequisite: K1–K15 · Reading: reference — use as needed · Difficulty: all levels