Problem 50 - University Calculus: Early Transcendentals

Question

Airplane landing path 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 $$y(x)=H\left[2\left(\frac{x}{L}\right)^{3}+3\left(\frac{x}{L}\right)^{2}\right]$$ GRAPH CANT COPY

Step-by-Step Solution

Answer

The answer is zero for both derivative conditions.

Step 1: Set up the polynomial function and its derivatives

Let's start with the polynomial function given:\[ y(x) = ax^3 + bx^2 + cx + d \]The derivative of this function with respect to \( x \) is:\[ \frac{dy}{dx} = 3ax^2 + 2bx + c \]

Step 2: Apply the boundary condition for y(x)

According to the problem, the airplane must be at altitude \( H \) when the horizontal distance is \(-L\), and at altitude 0 when the distance is 0. So we have:1. \( y(-L) = H \) which gives:\[ a(-L)^3 + b(-L)^2 + c(-L) + d = H \]2. \( y(0) = 0 \) which gives:\[ d = 0 \]Substituting \( d = 0 \) into the first equation:\[ -aL^3 + bL^2 - cL = H \]

Step 3: Determine derivative condition at x=0

The question asks for the derivative \( \frac{dy}{dx} \) at \( x = 0 \):\[ \frac{dy}{dx} \Big|_{x=0} = c = 0 \]

Step 4: Determine derivative condition at x=-L

The question asks for the derivative \( \frac{dy}{dx} \) at \( x = -L \), and it is given that the slope at the beginning of the descent is 0 (horizontal glide path):\[ \frac{dy}{dx} \Big|_{x=-L} = 0 \]Substitute \( x = -L \) into the derivative:\[ 3a(-L)^2 + 2b(-L) + c = 0 \]Since \( c = 0 \), this simplifies to:\[ 3aL^2 - 2bL = 0 \]

Step 5: Solve equations to find coefficients a and b

From both boundary conditions, solve for \( a \) and \( b \):1. \( -aL^3 + bL^2 = H \)2. \( 3aL^2 - 2bL = 0 \)Solving the second equation for \( b \):\[ b = \frac{3aL}{2} \]Substitute into the first equation:\[ -aL^3 + \frac{3aL}{2}L^2 = H \]\[ -aL^3 + \frac{3aL^3}{2} = H \]\[ \frac{aL^3}{2} = H \]\[ a = \frac{2H}{L^3} \]Substitute \( a \) back to find \( b \):\[ b = \frac{3}{2} \left(\frac{2H}{L^3}\right)L = \frac{3H}{L^2} \]

Step 6: Write final polynomial and verify

Substitute values of \( a \) and \( b \) into the form \( y(x) = ax^3 + bx^2 \) to obtain:\[ y(x) = \frac{2H}{L^3}x^3 + \frac{3H}{L^2}x^2 \]Factor out \( H \) and adjust terms:\[ y(x) = H \left( \frac{2x^3}{L^3} + \frac{3x^2}{L^2} \right) = H \left[ 2\left(\frac{x}{L}\right)^3 + 3\left(\frac{x}{L}\right)^2\right] \]This matches the expected form given in the problem statement.

Key Concepts

Derivative

The derivative of a function gives us the rate at which the function's value changes with respect to a change in a variable. In the context of a cubic polynomial function like this problem's airplane landing path, the derivative is crucial to understanding the slope or steepness at any point along the path. For a cubic polynomial function given by \( y(x) = ax^3 + bx^2 + cx + d \), the derivative, denoted as \( \frac{dy}{dx} \), is found by applying basic rules of calculus. Each term in the function is differentiated:

• The term \( ax^3 \) becomes \( 3ax^2 \)
• The term \( bx^2 \) becomes \( 2bx \)
• The term \( cx \) becomes \( c \)
• The constant \( d \) disappears as its rate of change is zero

Hence, the derivative is: \[ \frac{dy}{dx} = 3ax^2 + 2bx + c \] This formula tells us how steep the path is, which is essential for adjusting safely during the airplane's descent.

Boundary Conditions

In this exercise, boundary conditions are crucial for determining specific coefficients of the polynomial. These conditions are essentially constraints that the solution must satisfy. Here, two main boundary conditions are given: the airplane's altitude when it starts its descent, and its altitude when it reaches the runway.

• The first condition specifies that at the start of the descent, the airplane is at altitude \( H \). Mathematically, this is expressed as \( y(-L) = H \).
• The second condition specifies that at the end of the descent, the airplane is on the runway, or at altitude 0. This means \( y(0) = 0 \).

These boundary conditions allow us to form equations with the polynomial's coefficients, helping us find the exact form of the descent path.

Coefficient Determination

Once the boundary conditions and derivatives are applied, we can determine the coefficients in the polynomial. We use the information that \( y(-L) = H \) and \( y(0) = 0 \), along with the derivative conditions at these points, to form a system of equations. By solving these:

• From \( y(-L) = H \), we substitute into the polynomial to get \( -aL^3 + bL^2 - cL = H \)
• From \( y(0) = 0 \), we find \( d = 0 \)
• Using the derivative conditions,\( c = 0 \) at \( x=0 \), simplifying our equations further
• The slope condition at \( x = -L \) provides another equation: \( 3aL^2 - 2bL = 0 \)

By solving these equations for \( a \) and \( b \), we find \( a = \frac{2H}{L^3} \) and \( b = \frac{3H}{L^2} \), thus determining the specific form of our polynomial path.

Polynomial Derivatives

Polynomial derivatives give insight into how the polynomial function behaves, especially its rate of change at various points. In landing path problems, understanding polynomial derivatives can ensure safe and smooth trajectories for aircraft. For a cubic polynomial function, the derivative helps us compute slopes, which are vital for maintaining safe descent angles.

• The first derivative, \( \frac{dy}{dx} = 3ax^2 + 2bx + c \), indicates the path's slope. Knowing the slope at the beginning (\( x = -L \)) and end (\( x = 0 \)) helps pilots control the glide path.
• A zero derivative (flat slope) at certain points can indicate crucial areas where adjustments are necessary.

Mastering polynomial derivatives aids in solving real-world engineering and physics problems like airplane descents, ensuring safety and precision in navigation.