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Newtonian Mechanics: Kinematics, Newton's Laws, and the Forces

Part 1 of 5 of the Newtonian Mechanics reference (plugin newtonian-mechanics), covering §0–§3. Sibling skills: mech-energy-momentum-and-collisions (§4–§5), mech-rotation-rigid-bodies-and-oscillations (§6–§8), mech-orbits-frames-analytical-mechanics-and-simulation (§9–§14), mech-reference (§15–§19). Section numbers are shared across the set; a reference written as §N → skill points into that sibling skill.

Currency: Settled since 1687 — Newton's Principia, Euler's rigid-body work in the 1750s, Lagrange 1788, Hamilton 1833. Nothing here has changed or will.

Scope. Complements a fundamental-physics reference, which covers relativity, quantum mechanics and where classical mechanics breaks down. ⚠️ This is the classical theory done properly, including §14 → mech-orbits-frames-analytical-mechanics-and-simulation on numerical integration — the part that matters if you're simulating any of it.

⚠️ GOTCHA boxes mark genuine misconceptions, including several that survive a physics degree.

The three ideas that reorganize everything once you see them:

1. ⚠️ Force causes acceleration, not velocity. This one sentence is the entire content of the Aristotle-to-Newton revolution, and the misconception it replaced is the single most robust error in physics education (§2.2).
2. ⚠️ The conservation laws are not consequences of F = ma — they're deeper than it. Momentum conservation follows from spatial translation symmetry, energy from time-translation symmetry, angular momentum from rotational symmetry (Noether, 1918). They survive into relativity and quantum mechanics; F = ma does not (§4.5 → mech-energy-momentum-and-collisions).
3. ⚠️ Newtonian mechanics is deterministic but not predictable. Determinism is a property of the equations; predictability is a property of your knowledge. Chaos separates them, and the separation is not a defect of your instruments (§13 → mech-orbits-frames-analytical-mechanics-and-simulation).

§0. Routing

§1. Kinematics

Description of motion, before any mention of cause.

v = dr/dt        a = dv/dt = d²r/dt²
Constant acceleration only:
  v = v₀ + at        r = r₀ + v₀t + ½at²        v² = v₀² + 2a·Δr

⚠️ GOTCHA — those three equations are valid only for constant acceleration, and students apply them everywhere. The moment a depends on position (a spring), on velocity (drag), or on time, they are wrong. ⚠️ The general case requires integrating the differential equation — which is why §14 → mech-orbits-frames-analytical-mechanics-and-simulation exists.

⚠️ Velocity and acceleration are independent. A body can have zero velocity and nonzero acceleration (a ball at the top of its arc — this is the classic exam question), or constant speed and nonzero acceleration (uniform circular motion).

Circular motion: a_c = v²/r = ω²r, directed toward the centre. Angular quantities θ, ω, α mirror the linear ones exactly.

Projectile motion: horizontal and vertical decouple (⚠️ without drag — with drag they couple, and there is no closed-form solution; see §3.4). Range on level ground is v₀² sin(2θ)/g, maximum at 45°.

⚠️ Relative motion: v_AC = v_AB + v_BC. Galilean velocity addition, and it's exactly what special relativity replaces.

§2. Newton's Laws

2.1 Stated precisely

First law — a body remains at rest or in uniform straight-line motion unless acted on by a net external force.

⚠️ GOTCHA — the first law is not a special case of the second. It looks redundant (F=0 ⟹ a=0), but its real content is defining what an inertial frame is: a frame in which the first law holds. ⚠️ The second law is only valid in such frames, so the first law is the precondition for the second, not a corollary of it (§10 → mech-orbits-frames-analytical-mechanics-and-simulation).

Second law — ⚠️ properly F = dp/dt, not F = ma. They agree only when mass is constant. For variable-mass systems — a rocket, a falling chain, a conveyor being loaded — you must go back to momentum, and ⚠️ naively writing F = ma with m(t) is wrong, because it ignores the momentum carried by the mass entering or leaving. (See a rocket-science reference for the correct derivation.)

Third law — forces come in equal, opposite pairs on different bodies.

2.2 ⚠️ The misconceptions the laws generate

These are documented, robust, and survive instruction — the Force Concept Inventory literature exists because of them.

• ⚠️ "Motion requires a force." The most persistent error in physics. Constant velocity requires zero net force. ⚠️ It feels wrong because on Earth friction is always present, so maintaining motion does require force — to cancel friction, not to sustain the motion.
• ⚠️ "A thrown ball has a forward force on it." It does not. After release, the only forces are gravity and drag. The "force of the throw" is not a thing that persists — ⚠️ what persists is momentum, and confusing the two is the whole error.
• ⚠️ "The third-law pair cancels." ⚠️ They act on different bodies and therefore never cancel each other in a single body's free-body diagram. Cancellation would make all acceleration impossible.
• ⚠️ "A heavier object falls faster." Not in vacuum. In air, terminal velocity depends on the mass-to-drag ratio, so heavier usually does fall faster in practice — which is exactly why the misconception is so durable.
• ⚠️ "Centrifugal force pushes you outward." ⚠️ In an inertial frame there is no such force; you feel the seat pushing you inward (centripetal) and your inertia resisting. Centrifugal force is real and useful — but only in the rotating frame (§10 → mech-orbits-frames-analytical-mechanics-and-simulation).

2.3 Free-body diagrams

⚠️ The single most valuable procedural skill in mechanics, and it's mechanical:

1. Isolate ONE body. Draw it alone.
2. Draw ONLY forces acting ON it. ⚠️ Not forces it exerts. Not "the force of motion."
3. Every force must have an identifiable other object exerting it.
   ⚠️ If you can't name the exerter, the force isn't real.
4. Choose axes — align one with the acceleration if possible.
5. Write ΣF = ma per axis. Solve.

⚠️ Rule 3 eliminates nearly every spurious force students invent.

§3. The Forces

3.1 Gravity

F = Gm₁m₂/r², and near a surface W = mg. ⚠️ A spherically symmetric body attracts external objects as if all its mass were at the centre (Newton's shell theorem — and it took him years, which is a useful thing to know when it's presented as obvious). ⚠️ Inside a uniform shell, the field is exactly zero.

⚠️ Weight vs mass: mass is invariant; weight is mg and depends on where you are. "Weightlessness" in orbit is free fall, not absence of gravity — ⚠️ gravity at ISS altitude is about 90% of its surface value. The station and its occupants are accelerating together.

3.2 Normal force

⚠️ Perpendicular to the surface, and it is not equal to mg in general. It is whatever it needs to be to prevent interpenetration — on an incline, in an accelerating lift, or under an applied force, it differs. ⚠️ "N = mg" is a special case that students promote to a law.

3.3 Friction

Static:  f_s ≤ μ_s N     ⚠️ an INEQUALITY — it takes whatever value prevents sliding,
                         up to the maximum
Kinetic: f_k = μ_k N     ⚠️ roughly constant, opposing relative motion; μ_k < μ_s

⚠️ The inequality is the part people miss. A block at rest on a table with no applied force has zero friction, not μ_s N. You compute static friction from equilibrium, not from the formula — the formula only gives you the threshold.

⚠️ The Coulomb model's surprising claim: friction is independent of contact area, because real contact happens at asperities whose true contact area scales with normal force. ⚠️ It's an approximation — it fails for very soft materials, very clean surfaces (which can cold-weld), and at high speed. Racing tyres are wide for thermal and wear reasons the simple model doesn't capture.

Rolling resistance is a different mechanism entirely — hysteretic deformation loss, not sliding — and is much smaller.

3.4 Drag

Low Reynolds number (viscous):   F = −bv        ⚠️ linear
High Reynolds number (inertial): F = ½ρCdAv²    ⚠️ quadratic — the everyday case

⚠️ Terminal velocity when drag balances weight: v_t = √(2mg/ρC_dA). ⚠️ Quadratic drag makes the equations non-integrable in closed form for most cases — which is precisely why projectile problems in textbooks ignore it and why real ballistics is numerical.

3.5 Spring and tension

Hooke's law F = −kx — ⚠️ linear only within the elastic limit, and the minus sign is the physics: the force opposes displacement, which is what makes oscillation possible. Tension — ⚠️ uniform throughout an ideal massless string, and an ideal pulley changes tension's direction without changing its magnitude. Real ropes have mass and real pulleys have inertia and friction.