A claim-first course

Unit M1 · Falling & bouncing

Infinitely Many Bounces · objectives: restitution as a speed fraction · energy goes as speed squared · a convergent geometric series of bounces

1. Set h₀ = 3, e = 0.6. Predict the third apex height before looking, then check.
   Answer

   h₃ = e⁶·h₀ = 0.6⁶·3 = 0.140 m. If you predicted e³ instead of e⁶, you conflated speed with energy — the most common miss in this unit.

2. Find, by algebra first and slider second, the e that makes a 4 m drop rest at exactly 10 s on Earth.
   Answer

   Solve √(2h₀/g)·(1+e)/(1−e) = 10 → (1+e)/(1−e) = 11.07 → e ≈ 0.834.

3. On the Moon slider, which readouts change and which are invariant? State the rule before touching it.
   Answer

   All times scale as 1/√g; every apex height is invariant — hₙ = e²ₙh₀ has no g in it.

Unit M2 · One fall, three mechanics

One Fall, Three Mechanics · objectives: F = ma, δS = 0 and Hamiltonian flow as one motion · conservation and exactly where it ends

1. Pause anywhere in flight. From the H readout alone, predict the next apex height, then scrub to it.
   Answer

   At the apex all of H is potential: h = H/g (unit mass).

2. Read H just after two successive bounces and recover e from the pair.
   Answer

   e = √(Hₙ₊₁/Hₙ) — the staircase is a measurement instrument.

3. Why is the action curve symmetric in ε? What would an asymmetric curve tell you about the flown path?
   Answer

   The linear term of S(ε) is the Euler–Lagrange expression integrated against the perturbation; it vanishes because the flown path obeys Newton. An asymmetric (linear) term would mean the reference path was not a true trajectory.

4. Break it (Rust rung): in claim C4, change e * e to e. Predict the failing test and the measured ratio it will print, then run cargo test -p lesson-three-mechanics.

Unit M3 · The projectile (and the template)

Projectile Range · objectives: the τ/8 maximum · complementary angles · natural units and scale invariance

1. Two angles give range 70% of best. Find both on the slider; verify their sum is τ/4 (claim P3), then prove it from sin(2θ).
2. Fix θ. How must range and flight time each scale when g quadruples? Predict, then use the slider.
   Answer

   Both go as 1/g at fixed v: range v²sin2θ/g, time 2v sinθ/g. (Apex too.)

3. After the authoring guide: add a claim P6 asserting the flight time formula against a sampled crossing, and make it pass.

Unit W1 · Waves from Newton

Deriving the Wave Equation · objectives: second difference as curvature · the sin→tan budget · dispersion and the continuum limit · superposition

1. Pluck at 30%. Predict when the left-going pulse reaches the wall at c = 1.5, then scrub to it.
   Answer

   Distance 3 at speed 1.5 → t = 2.0 s.

2. Using step 2's error readout, find the amplitude where tan-for-sin crosses 5%. Then derive the small-slope law error ≈ slope²/2 and check it explains your number.
3. Pluck dead centre and go to step 3: one bead never feels a net pull. Which, and what symmetry protects it?
   Answer

   The centre bead: its two neighbours mirror each other, so the second difference vanishes there for every symmetric mode the pluck excites.

4. At N = 5, describe what trails the main pulse and why raising N — which never fixes the lattice cutoff (claim W4) — still cleans it up.
   Answer

   Dispersion: short-wavelength modes lag (their speed pins at 4/τ of c). A smooth pluck puts ever less energy into them as N grows — that is exactly what the fidelity readout measures.

Unit GA1 · Two mirrors, one rotor

Two Mirrors Make a Rotation · objectives: reflection as a sandwich · rotors as reflection pairs · angle doubling · independence from the rotated vector

1. Predict the mirror separation that turns v by exactly one third of a turn, then set it.
   Answer

   Δ = 1/6 τ — half the rotation, always.

2. Beyond Δ = 0.25 τ the readout ⟨R⟩₀ goes negative. What does a negative scalar part mean for the rotor, and does the rotation care?
   Answer

   ⟨R⟩₀ = cosΔ < 0 past a quarter turn; R and −R implement the same rotation (the double cover) — the motion is oblivious.

3. Thought experiment the page can't do yet: add a third mirror. Is the composite a rotation? What single operation must it be, and why does the count's parity decide?
   Answer

   Odd count → orientation reverses → it is a reflection (about some line), never a rotation. Parity of reflections is the deepest invariant in the unit.

4. Break it: in C3, flip the expected sign of the e₁₂ component and predict the exact failure grid before running the tests.

Unit O1 · Images, real and virtual

Real or Virtual · objectives: the five converging cases · the sign of f as the true category · where real images can be caught

1. With f = 2, find the object distance where the image is real and exactly life-size, for the lens and for the concave mirror. Compare.
   Answer

   dₒ = 2f = 4 in both — same equation, same case structure; only the side the image lands on differs.

2. Why can a convex security mirror never lie about orientation? Answer from the case table, not the picture.
3. Sweep through dₒ = f slowly and describe what happens to the image on both sides of the crossing.

For instructors. Every exercise above is anchored to a claim with a named test, so grading can be mechanical where you want it: a student's "break it" homework is a diff plus the failing test's output. The lab re-proves all claims in public CI on every push; fork the repo and your students' pull requests get the same treatment. To assign new exercises, add tryThis entries to a lesson's manifest — or have students build whole lessons: that is the authoring guide.