Problem 50 - Algebra 2

Question

AEROSPACE NASA's KC135 A aircraft flies in parabolic arcs to simulate the weightlessness experienced by astronauts in space. The height h of the aircraft (in feet) \(t\) seconds after it begins its parabolic flight can be modeled by the equation \(h(t)=-9.09(t-32.5)^{2}+34,000 .\) What is the maximum height of the aircraft during this maneuver and when does it occur?

Step-by-Step Solution

Answer

The maximum height is 34,000 feet at 32.5 seconds.

Step 1: Understanding the Equation

The given quadratic equation modeling the height of the aircraft is \( h(t) = -9.09(t-32.5)^2 + 34,000 \). This equation is in the form of \( a(x - h)^2 + k \), which represents a downward-opening parabola. The maximum point or vertex of this parabola gives the maximum height the aircraft reaches.

Step 2: Identifying the Vertex

For the equation \( h(t) = -9.09(t-32.5)^2 + 34,000 \), the vertex form \( (t-h)^2 \) means the vertex occurs at \( t = 32.5 \). This is the time in seconds when the aircraft reaches its maximum height. The height at this time is given by the remaining part of the equation.

Step 3: Calculating the Maximum Height

Since the equation is already in vertex form, the maximum height can be directly read from the equation. At \( t = 32.5 \), the maximum height \( h(32.5) = 34,000 \) feet. Thus, the aircraft reaches its peak height at this point.

Key Concepts

Parabolic Motion

Parabolic motion is a type of movement that creates a symmetric curve known as a parabola. This curve is common in scenarios where an object moves through space under the influence of gravity.
For example, when you throw a ball, it arcs up and then falls back down, creating a parabolic path. This smooth, symmetric arc is due to the force of gravity acting on the object, pulling it downwards as it moves forward.
Understanding parabolic motion is crucial in aerospace science, especially for simulating weightlessness. The arc or parabola allows scientists like those at NASA to mimic the conditions astronauts experience in space.

Vertex Form

Vertex form is a specific way of writing quadratic equations that highlights the vertex of the parabola. It is structured as:

• \[ y = a(x-h)^2 + k \]

Here,

• \((h, k)\) is the vertex of the parabola, representing the maximum or minimum point of the graph.
• \(a\) determines whether the parabola opens upwards or downwards.

If \( a < 0 \), the parabola opens downward, indicating a maximum point, such as with the aircraft's flight curve described in the exercise.
Vertex form is incredibly useful because it provides important information at a glance, including both the highest point and the time at which it occurs.

Maximum Height

The maximum height in a quadratic function refers to the peak point of the parabolic curve. For the NASA aircraft, this was given by the vertex \((t, h(t))\) of the equation.
In the equation used to describe the aircraft's path:

• \(t = 32.5\)
• \(h(t) = 34000\) feet

This means the aircraft reaches its highest point, 34,000 feet, 32.5 seconds into its parabolic flight.
Identifying the maximum height is crucial, especially in aerospace simulations, to ensure the safety and effectiveness of the maneuvers.

NASA Aerospace Simulations

NASA uses parabolic arcs in their aerospace simulations to recreate a weightless environment. This technique is vital for training astronauts who will experience microgravity conditions in space.
By flying aircraft in parabolic paths, NASA can provide brief periods of weightlessness. During these arcs, objects inside the aircraft appear to "float," similar to the conditions astronauts face.
These simulations are an essential part of preparing for missions, allowing astronauts to acclimatize to microgravity before entering space. Understanding the physics behind these arcs, including equations and maximum height, ensures accurate and reliable simulations.