Beam on the Rocks — Print Cards

Teacher — unlock codes (cut off before distributing student cards)

Phase 1: SWEEP

Phase 2: CHAINR

Phase 3: TURTLE

Phase 4: OUTRUN

🔦 The Scenario

A lighthouse, a rotating beam, and a straight stretch of coast

A lighthouse rotates its beam at a constant angular speed, lighting up a single spot as it crosses a long, straight shore. Your job: find exactly how fast that spot moves, and whether "constant turning speed" means "constant sliding speed." It doesn't.

Watch the intro animation on the class screen before Task 1a, then keep this card at your whiteboard, you'll use it across all four phases.

⏱ ~55 min  ·  👥 Groups of 3–4  ·  HL only

Phase 1 · The Hook & the Guess

Setting up the model

Task 1a

Sketch it, then commit to a guess

Draw the lighthouse, the shore, and the beam at three moments. Does the spot slide at a constant speed? Commit to yes or no before any maths.

Task 1b

Build the equation

Label \(d\) (perpendicular distance), \(\theta\) (angle from perpendicular), \(x\) (distance along the shore). Use right-triangle trig to connect them.

Task 1c

Stress-test your equation

What does it give at \(\theta = 0\)? What happens as \(\theta \to 90°\)? Keep this for Phase 3.

Phase 2 · Chasing a Moving Variable

Implicit differentiation in \(t\)

Task 2a

\(\dfrac{d}{dt}(x^2)\)

Same chain rule you already use for implicit differentiation, just with respect to \(t\) instead of \(x\).

Task 2b

\(\dfrac{d}{dt}(\tan\theta)\)

Recall the derivative of \(\tan\theta\), then apply the same chain rule.

Task 2c — the big one

Differentiate your own equation

Apply 2a and 2b to your Task 1b equation. This is the related rates equation for the whole activity.

Phase 3 · The First Sweep

Evaluating at an instant

Task 3a

Rotation period → angular speed

One rotation every \(40\pi\) s. Show this gives \(\dfrac{d\theta}{dt} = 0.05\) rad/s.

Task 3b

Evaluate at three angles

With \(d = 200\): find \(\dfrac{dx}{dt}\) at \(\theta = 0°, 30°, 60°\). Exact fractions, then decimals.

Task 3c

Back to your prediction

Were you right? Explain why, using \(\sec^2\theta\).

Task 3d — Challenge

Faster than sound

Find \(\theta\) where the spot hits 343 m/s. What's actually moving that fast, and what isn't?

Phase 4 · Outrunning the Beam

Solving for an unknown rate

Task 4a

Find the critical angle

A researcher walks the shore at \(v = 15\) m/s. Find \(\theta\) where the beam matches her speed.

Task 4b

Find the position

At that angle, find \(x\) exactly (surd form), then as a decimal.

Task 4c — the real question

Can she outwalk it?

Before and after the critical angle. Is she eventually caught, no matter how she paces herself? Justify with how \(\sec^2\theta\) behaves.