Marius Hills · candidate skylightGround-penetrating radarONE relationship: d = v·t/2
🎯 Learning target
Master ONE relationship — echo ranging: d = v·t/2 (depth = signal speed × round-trip time ÷ 2) Forward, inverse, and under a cave-in clock.
⚠️ Misconception to break
"The echo time IS the depth." No — the pulse travels down AND back. The depth is HALF of what the round trip suggests. The ÷2 is the whole game.
🧪 Model & assumptions (stated honestly)
Signal speed in a medium is v = c/√εᵣ; here v is given per material and the graded law is d = v·t/2 (permittivity εᵣ is the "why", not graded) · working units: d in metres, v in ×10⁸ m/s, t in microseconds (µs), so d = 50·ṽ·t · one flat reflector per ping, normal incidence · the real GPR trade-off (frequency ↔ penetration ↔ resolution) is acknowledged in this briefing but never graded · teaching εᵣ rounded to clean (c/v)².
Difficulty generous tolerances
📐 Key formulas (given)
echo ranging: d = v · t / 2
working form: d[m] = 50 · ṽ · t
ṽ = v / 10⁸ (m/s) · t in µs
inverse gate: t = 2d / v
medium speed: v = 2d / t (then v = c/√εᵣ)
📡 Curriculum — 4 surveys
1️⃣ First Pings · Watch — echo time is linear in depth 2️⃣ Sounding · Predict — v & t given; your depth arms the log 3️⃣ Range Gate · Inverse — target a depth; set the echo window 4️⃣ Cave-In · Crisis — 90 s: name the layer, clear the void ceiling Bronze+ opens the next survey. Scenarios re-roll every retry.
Controls
Space ping/launch · Enter commit · P pause · R reset · H hint. Each survey opens with a 🧠 reasoning question. Wrong submissions −2 pts.
🏅 How scoring works
Every stage: answer the reasoning question, then compute and commit a number. Your medal is set by accuracy — 🥇 under 5% · 🥈 under 10% · 🥉 under 20% error — and a Bronze or better unlocks the next stage. A wrong answer costs −2 points (−1 on Easy, and your first reasoning slip on each question is free on Easy). A hint costs −3/−4/−5. Fewer tries and higher accuracy mean a higher score — so it always pays to calculate, not guess.