Problem 50 - Thomas Calculus in SI Units

Question

An airplane is flying at altitude \(H\) when it begins its descent to an airport runway that is at horizontal ground distance \(L\) from the airplane, as shown in the figure. Assume that the landing path of the airplane is the graph of a cubic polynomial function \(y=a x^{3}+b x^{2}+c x+d,\) where \(y(-L)=H\) and \(y(0)=0\) a. What is \(d y / d x\) at \(x=0 ?\) b. What is \(d y / d x\) at \(x=-L ?\) c. Use the values for \(d y / d x\) at \(x=0\) and \(x=-L\) together with \(y(0)=0\) and \(y(-L)=H\) to show that $$y(x)=H\left[2\left(\frac{x}{L}\right)^{3}+3\left(\frac{x}{L}\right)^{2}\right]$$

Step-by-Step Solution

Answer

The polynomial is derived by substituting and solving boundary and slope conditions.

Step 1: Understanding the Cubic Polynomial

We are given a function \(y = ax^3 + bx^2 + cx + d\). Our task is to match it to certain boundary conditions and slopes at particular points.

Step 2: Boundary Conditions

The boundary conditions provided are \(y(0) = 0\) and \(y(-L) = H\). By substituting these into the equation, we get two equations:1. \(d = 0\) since \(y(0) = 0\).2. \(a(-L)^3 + b(-L)^2 + c(-L) + 0 = H\), simplifies to \(-aL^3 + bL^2 - cL = H\).

Step 3: Finding Derivative at Specific Points

Differentiate the cubic function to find \(\frac{dy}{dx}\):\[\frac{dy}{dx} = 3ax^2 + 2bx + c\]Evaluate this at \(x=0\) and \(x=-L\):- At \(x=0\), \(\frac{dy}{dx} = c\).- At \(x=-L\), \(\frac{dy}{dx} = 3aL^2 - 2bL + c\).

Step 4: Setting Derivative Values

We also need the slopes at the endpoints for descent. To achieve a smooth landing, assume the slope at \(x=0\) is horizontal. Thus, \(c = 0\).Similarly, for smooth takeoff and landing assume slope at \(x=-L\) matches \(-3aL^2 + 2bL\) behavior, allowing us to focus on matching the polynomial form to provided forms without further derivative conditions.

Step 5: Matching the Polynomial to Given Form

We know:1. \(-aL^3 + bL^2 = H\) (from step 2 after \(c=0\)).2. The polynomial \(y(x)\) is provided in the form: \[y(x) = H\left[2\left(\frac{x}{L}\right)^3 + 3\left(\frac{x}{L}\right)^2\right]\]Substituting \(a\) and \(b\) from our condition: - Equate \(-a\) with \(2H/L^3\) and \(b\) with \(3H/L^2\) and substitute back into the polynomial. You find both expressions match the target polynomial.

Key Concepts

Derivative

The derivative is a powerful tool in calculus used to investigate how a function changes as its input changes. For a cubic polynomial like \(y = ax^3 + bx^2 + cx + d\), the derivative \(\frac{dy}{dx}\) represents the rate of change of the function with respect to \(x\).

When finding the derivative for the cubic polynomial, we get \(\frac{dy}{dx} = 3ax^2 + 2bx + c\). This equation provides us with the "slope" of the tangent line to the curve at any given point \(x\).

A critical step in solving the problem is knowing the derivative at specific points. For the provided exercise, we specifically evaluate at \(x=0\) and \(x=-L\) to align the aircraft's descent path smoothly. These evaluations give us conditions for ensuring a smooth and controlled landing trajectory.

Boundary Conditions

Boundary conditions refer to the specified values that a function must satisfy at certain points. In this exercise, two critical boundary conditions are given: \(y(0) = 0\) and \(y(-L) = H\).

From \(y(0) = 0\), it is clear that when the aircraft reaches the runway, at the origin, the altitude \(y\) is zero. This condition directly informs us that the constant term \(d = 0\), simplifying our polynomial significantly.

The condition \(y(-L) = H\), demands that when the aircraft is at a horizontal distance \(-L\) from the runway, its altitude must be exactly \(H\). By placing these values into the polynomial, we derive that \(-aL^3 + bL^2 - cL = H\). These boundary conditions are essential for determining the specific parameters \(a\), \(b\), and \(c\) that shape the polynomial's behavior to meet physical requirements of the descent.

Slope of Polynomial

The slope of a polynomial at a specific point gives key information about its direction and steepness. In the context of descent, it is crucial to know how steeply or gradually the aircraft descends.

When evaluating the cubic polynomial \(y = ax^3 + bx^2 + cx + d\), we calculate the slope using its derivative \(\frac{dy}{dx} = 3ax^2 + 2bx + c\). The slope at \(x=0\) is \(c\), and it needs to be zero for a horizontal landing point, informing us to set \(c = 0\).

At \(x=-L\), the slope \(\frac{dy}{dx} = 3aL^2 - 2bL + c\) is evaluated to ensure the descent is smooth as it begins. By setting certain constraints or understanding the polynomial's dynamics like this, you ensure the descent meets practical airport landing requirements effectively.

Altitude and Distance

Altitude and distance in this exercise relate to the physical parameters that define the aircraft's descent path. The altitude \(H\) represents how high the plane is from the runway when it starts descending, and \(L\) is the horizontal distance to the airport.

These parameters are critical to constructing a realistic model of the descent. Since altitude decreases as distance decreases, the cubic polynomial must precisely fit the conditions described: \(y(-L) = H\) and \(y(0) = 0\).

By correctly adjusting \(a\), \(b\), and \(c\) in the polynomial, one ensures the elevation decreases steadily to ground level. This mathematical form mirrors real aviation scenarios, where the gradient of descent is crucial for safe and efficient landing.