Beam on the Rocks

You already know implicit differentiation: if \(y\) is secretly a function of \(x\), then \(\dfrac{d}{dx}(y^2) = 2y\dfrac{dy}{dx}\), by the chain rule.

Here's the twist. Suppose instead that \(x\) is secretly a function of time, \(t\), even though you don't know its formula. Using exactly the same reasoning: what is \(\dfrac{d}{dt}(x^2)\)?

Suppose \(\theta\) is also a function of \(t\). Using the same chain rule, and recalling the derivative of \(\tan\theta\) with respect to \(\theta\), find \(\dfrac{d}{dt}(\tan\theta)\).

Go back to your equation from Task 1b. In this problem, \(d\) is fixed, but \(x\) and \(\theta\) both change as time passes; both are secretly functions of \(t\).

Differentiate both sides of your equation with respect to \(t\). Use Tasks 2a and 2b as your model. Write the result on your board: this is the equation that connects how fast the beam turns to how fast the spot slides.

The lighthouse sits \(d = 200\) m from the shore. Its beam completes one full rotation every \(40\pi\) seconds (about 126 s).

One full rotation is \(2\pi\) radians. Show that a period of \(40\pi\) seconds corresponds to an angular speed of \(\dfrac{d\theta}{dt} = 0.05\) rad/s.

Using \(d = 200\) and \(\dfrac{d\theta}{dt} = 0.05\), find \(\dfrac{dx}{dt}\), the speed of the spot along the shore, at each of these three moments:

Show full working for each. Give exact fractions where you can, then a decimal in m/s.

Look at your three answers to Task 3b. Was your Phase 1 guess correct? In your own words, explain why a beam turning at a perfectly constant angular speed does not produce a spot sliding at a constant linear speed. Use the word "\(\sec^2\theta\)" in your explanation.

Sound travels at roughly 343 m/s. Using your Task 2c equation, find the angle \(\theta\) at which the spot of light would be sliding along the shore at exactly this speed.

This is a genuine answer, not a trick, but nothing physical is actually travelling that fast. What is really moving at the lighthouse, and what is just appearing to move along the shore? Why doesn't this break any law of physics?

A marine biology field researcher is walking along the waterline of the same beach every night, checking a line of sea turtle nests. Nesting turtles are easily startled by direct light, so the researcher tries to always stay ahead of the beam's spot rather than get caught in it. Tonight she is walking at a steady \(v = 15\) m/s along the shore.

Using the same lighthouse (\(d = 200\), \(\dfrac{d\theta}{dt} = 0.05\) rad/s), find the angle \(\theta\) at which the spot of light is sliding along the shore at exactly \(v = 15\) m/s, the same speed as the researcher. Set up the equation from Task 2c equal to \(15\) and solve for \(\theta\). Give an exact value if you can, then a decimal in degrees.

At that same angle, how far along the shore, \(x\), is the spot of light from the base point? Give an exact surd value, then a decimal in metres.

For angles smaller than your answer to 4a, is the beam's spot moving slower or faster than the researcher? What about for larger angles?

Once the beam has swept past your critical angle, could the researcher escape it by simply walking a little faster, or is she guaranteed to eventually be caught in it if she keeps checking nests along this stretch of coast for long enough? Justify your answer using how \(\sec^2\theta\) behaves as \(\theta \to 90°\), not just a gut feeling.