Problem 69
Question
Are based on the idea of a glide slope (the angle the flight path makes with the ground). Precision Approach Path Indicator (PAPI) lights are used as a visual approach slope aid for pilots landing aircraft. A typical glide path for commercial jet airliners is \(3^{\circ} .\) The space shuttle has an outer glide approach of \(18^{\circ}-20^{\circ} .\) PAPI lights are typically configured as a row of four lights. All four lights are on, but in different combinations of red or white. If all four lights are white, then the angle of descent is too high; if all four lights are red, then the angle of descent is too low; and if there are two white and two red, then the approach is perfect. (IMAGE CANNOT COPY) If a commercial jetliner is 5000 feet (about 1 mile) ground distance from the runway, what should the altitude of the plane be to achieve two red and two white PAPI lights? (Assume this corresponds to a \(\left.3^{\circ} \text { glide path. }\right)\)
Step-by-Step Solution
Verified Answer
The plane should be at an altitude of 262 feet.
1Step 1: Understanding Glide Path and Angle
A glide path is the angle a flight path makes with the ground. Given that for commercial jetliners this angle is typically 3°, to ensure a proper landing alignment.
2Step 2: Calculate Altitude Using Trigonometry
To find the altitude, use the tangent of the glide path angle. The formula for the tangent of an angle in a right triangle is: \[\tan(\theta) = \frac{\text{opposite side}}{\text{adjacent side}}\]Here, \(\theta = 3°\), the opposite side is the altitude we want to find, and the adjacent side is the ground distance from the runway (5000 feet).
3Step 3: Apply the Tan Function
Substitute the known values into the tangent formula:\[\tan(3^ 0) = \frac{\text{Altitude}}{5000}\]Calculate the tangent of 3° using a calculator: \(\tan(3°) \approx 0.0524\).
4Step 4: Solve for Altitude
Rearrange the equation to solve for altitude:\[\text{Altitude} = 5000 \times 0.0524 = 262\]
5Step 5: Conclusion
The altitude at which the plane should be to achieve two red and two white PAPI lights just before landing is 262 feet.
Key Concepts
Glide SlopeTrigonometryAngle of Descent
Glide Slope
Glide slope is an essential concept for pilots as it refers to the optimal angle of descent that an aircraft should maintain during its approach to a runway. Precision Approach Path Indicator (PAPI) lights are critical in visualizing this angle for pilots as they come in to land. When an aircraft is on the correct glide slope, pilots will see two white and two red lights from the PAPI system. This consistent coloration indicates that the aircraft is following the ideal descent angle, which for many commercial jetliners is usually set at 3 degrees. This optimal angle helps ensure a smooth and safe landing. It is important for pilots to monitor these lights carefully to adjust their flight path if necessary to maintain the correct glide slope.
Glide slope not only aids in keeping the plane level but also ensures it descends gradually enough to cover the ground distance safely without a steep approach. Adjustments may be necessary if the angle is too steep or too shallow, as indicated by four white or four red lights respectively. Thus, understanding and maintaining the glide slope is vital for landing safely.
Glide slope not only aids in keeping the plane level but also ensures it descends gradually enough to cover the ground distance safely without a steep approach. Adjustments may be necessary if the angle is too steep or too shallow, as indicated by four white or four red lights respectively. Thus, understanding and maintaining the glide slope is vital for landing safely.
Trigonometry
Trigonometry is a powerful mathematical tool that pilots use to calculate distances and angles during a flight. It involves the relationships between the angles and sides of triangles. In the context of aviation, trigonometry is used to compute the proper altitude of an aircraft as it follows the glide slope to the runway.
Using the basic trigonometric function of tangent, which relates the opposite and adjacent sides of a right triangle, pilots can determine the altitude required for a given slope. For instance, the formula:
Using the basic trigonometric function of tangent, which relates the opposite and adjacent sides of a right triangle, pilots can determine the altitude required for a given slope. For instance, the formula:
- \( \tan(\theta) = \frac{\text{Altitude}}{\text{Ground Distance}} \)
Angle of Descent
The angle of descent is a crucial element in the navigation and landing process of an aircraft. It describes the angle formed between the aircraft's flight path and the horizontal plane of the earth. An ideal descent angle, like the common 3 degrees for jetliners, balances safety and efficiency to bring airplanes in at a manageable steepness.
Monitoring the angle of descent is primarily achieved through PAPI lights, visually guiding pilots to maintain the correct course. When the angle is properly managed, aircraft minimize excessive fuel use and reduce the chances of a rough landing.
Maintaining a proper angle of descent encompasses not only compliance with visual guidance systems but also involves careful coordination with air traffic control to ensure approaches are synchronized with airport traffic. Achieving this balance involves a blend of physics, precise calculations, and experienced judgment, underscoring the importance of this concept in successful aircraft landings.
Monitoring the angle of descent is primarily achieved through PAPI lights, visually guiding pilots to maintain the correct course. When the angle is properly managed, aircraft minimize excessive fuel use and reduce the chances of a rough landing.
Maintaining a proper angle of descent encompasses not only compliance with visual guidance systems but also involves careful coordination with air traffic control to ensure approaches are synchronized with airport traffic. Achieving this balance involves a blend of physics, precise calculations, and experienced judgment, underscoring the importance of this concept in successful aircraft landings.
Other exercises in this chapter
Problem 69
In an oblique triangle \(A B C, \beta=45^{\circ}, \gamma=60^{\circ},\) and \(b=20\) in. Find the length of \(a\). Round your answer to the nearest unit.
View solution Problem 69
Find all possible values of \(\theta,\) where \(0^{\circ} \leq \theta \leq 360^{\circ}\) $$\cos \theta=\frac{\sqrt{3}}{2}$$
View solution Problem 69
Determine the angle of the smallest possible positive measure that is coterminal with each of the angles whose measure is given. Use degree or radian measures a
View solution Problem 70
In an oblique triangle \(A B C, \beta=\frac{2 \pi}{9}, \gamma=\frac{5 \pi}{9},\) and \(a=200 \mathrm{ft} .\) Find the length of \(c .\) Round your answer to the
View solution