Smooth Landing

At a glance

Pairs · launched with sinusoidal derivatives, due on the consolidation day · one descent, three demands, every derivative interviewed

What you are making

A regional jet begins its descent at an altitude of $10000$ m, twenty minutes out from the runway. You are designing the descent profile — the altitude function $h(t)$ — and it faces three demands: the plane must arrive (altitude zero exactly at the runway), it must touch gently (vertical speed as close to zero as you can manage at touchdown), and the ride must stay comfortable the whole way down (acceleration kept small, no sudden lurch). You finish with your chosen profile, its first and second derivatives, the three graphs aligned one above another, and a passenger report that says where the ride is gentlest, where it is worst, and what your profile still asks of the people in the seats.

Milestones

• Each demand translated into a precise statement about $h$, $h^{\prime }$, or $h^{\prime \prime }$ before any candidate profile is proposed
• Two candidate families differentiated with Derivative Rules and The Chain Rule — one polynomial, one sinusoidal
• Touchdown checked exactly: $h$ and $h^{\prime }$ evaluated at the moment of landing, not read off a zoomed graph
• Height, velocity, and acceleration graphed in a stack, per Motion on a Line, and the stories cross-checked
• The passenger report written, with every comfort claim traced to $h^{\prime \prime }$

How it is assessed

Per How Marks Work, the reasoning is the product: a profile that fails one demand, with the failure located and priced, outranks a perfect-looking graph nobody differentiated. On the due date your pair defends the touchdown out loud. The Math Journal entry on what the model ignores — wind, weight, the pilot — completes the evidence, with What Makes a Model Good as your guide.

Success criteria

Quality	What it looks like in your work
Demands made precise	Each demand written as mathematics first
A toolbox at work	Derivatives taken by rule and stated cleanly
Exact touchdown	$h$ and $h^{\prime }$ computed at landing, not eyeballed
Aligned evidence	Three stacked graphs tell one consistent story
An honest report	Comfort traced to $h^{\prime \prime }$, limits admitted

If your profile refuses one demand

Adjust one feature at a time — stretching the descent time makes everything gentler, at the price of arriving later. If gentleness and punctuality genuinely pull against each other in your design, that trade-off is the finding: price it and present it.

Curriculum connection

A3.5

solve problems, using the product and chain rules, involving the derivatives of polynomial functions, sinusoidal functions, exponential functions, rational functions [e.g., by expressing $f(x)=\frac{x^2+1}{x-1}$ as the product $f(x)=(x^2+1)(x-1{)}^{-1}$], radical functions [e.g., by expressing $f(x)=\sqrt{x^2+5}$ as the power $f(x)=(x^2+5{)}^{1/2}$], and other simple combinations of functions [e.g., $f(x)=x\sin x$, $f(x)=\frac{\sin x}{\cos x}$]*

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B2.1

make connections between the concept of motion (i.e., displacement, velocity, acceleration) and the concept of the derivative in a variety of ways (e.g., verbally, numerically, graphically, algebraically) Sample problem: Generate a displacement–time graph by walking in front of a motion sensor connected to a graphing calculator. Use your knowledge of derivatives to sketch the velocity–time and acceleration–time graphs. Verify the sketches by displaying the graphs on the graphing calculator.

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B2.5

solve problems arising from real-world applications by applying a mathematical model and the concepts and procedures associated with the derivative to determine mathematical results, and interpret and communicate the results Sample problem: A bird is foraging for berries. If it stays too long in any one patch it will be spending valuable foraging time looking for the hidden berries, but when it leaves it will have to spend time finding another patch. A model for the net amount of food energy in joules the bird gets if it spends $t$ minutes in a patch is $E=\frac{3000t}{t+4}$. Suppose the bird takes 2 min on average to find each new patch, and spends negligible energy doing so. How long should the bird spend in a patch to maximize its average rate of energy gain over the time spent flying to a patch and foraging in it? Use and compare numeric, graphical, and algebraic strategies to solve this problem.

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