Problem 71 - Precalculus

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 the pilot of the space shuttle Orbiter is at an altitude of 3000 feet when she is 15,500 ft (approximately 3 miles) from the shuttle landing facility (ground distance), what is her glide slope angle (round to the nearest degree)? Is she too high or too low?

Step-by-Step Solution

Answer

The glide slope angle is approximately 11°, and the pilot is too low.

Step 1: Define Key Variables

To find the glide slope angle, let's define the key variables: - Altitude (): 3,000 feet - Ground distance (): 15,500 feet

Step 2: Recall the Slope Formula

The glide slope angle  can be found using the tangent ratio for a right triangle, which is:\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\]Where the opposite side is the altitude and the adjacent side is the horizontal ground distance.

Step 3: Calculate the Tangent Value

Substitute the known values into the formula:\[\tan(\theta) = \frac{3000}{15500}\]Calculate the value:\[\tan(\theta) \approx 0.1935\]

Step 4: Determine the Angle

To find the angle , take the arctangent (inverse tangent) of the calculated tangent value:\[\theta \approx \arctan(0.1935)\]Using a calculator, we find:\[\theta \approx 11^{\circ}\]

Step 5: Assess the Approach

Compare the calculated glide slope angle, \(11^{\circ}\), with the typical space shuttle glide path range of \(18^{\circ}-20^{\circ}\). Since \(11^{\circ}\) is less than \(18^{\circ}\), the pilot is too low.

Key Concepts

Glide Slope

The glide slope is a crucial concept in aviation, representing the angle between an aircraft's flight path and the earth’s surface as it lands. It's an imaginary slope that a pilot follows to ensure a safe descent onto the runway. Pilots use precision instruments, like Precision Approach Path Indicator (PAPI) lights, to maintain the correct glide slope. These indicators help pilots assess whether they're on the right approach path.

In general aviation, a typical glide slope can be about 3 degrees for commercial jets, providing a gentle descent. However, for the space shuttle, a much steeper angle of 18 to 20 degrees is necessary due to its high speed and different landing dynamics. Maintaining the correct glide slope is essential for a safe landing, as deviations can indicate whether the aircraft is too high or too low, impacting the touchdown zone on the runway.

Tangent Ratio

The tangent ratio is a fundamental trigonometric concept. It relates the angles and lengths of a right triangle. Specifically, it's the ratio of the length of the side opposite the angle of interest to the length of the adjacent side. In mathematical terms, it's expressed as:

\( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \)

This ratio is especially useful in calculating angles when certain lengths are known, such as in aerial navigation. In the context of a glide slope, the tangent ratio helps determine the angle of descent by comparing the altitude (opposite) with the ground distance (adjacent). Understanding this concept allows pilots and engineers to design and execute precise landing approaches.

Arctangent Function

The arctangent function, also known as the inverse tangent, is used to determine the angle when you know the tangent value. If you already have the ratio from the tangent function, the arctangent helps in finding the specific angle associated with that ratio. It's denoted as \( \arctan(x) \) or sometimes \( \tan^{-1}(x) \).

In practical applications, such as calculating the glide slope angle in aviation, pilots use the arctangent function to interpret their current glide angle based on their altitude and ground distance. For example, if a pilot calculates a tangent value of 0.1935, using the arctangent will reveal the angle is approximately 11 degrees. Having access to this angle is vital for adjusting descent and ensuring a successful landing.

Angle of Descent

The angle of descent is a critical aspect of flight operations, particularly during landing. This angle is the actual angle at which an aircraft approaches the runway from the air. It's a measure of how steep or shallow the descent path is. A perfect angle of descent enables the aircraft to safely touch down within the designated area of the runway.

The angle is measured relative to the ground and can fluctuate based on aircraft type, approach speed, and weather conditions. PAPI lights provide visual cues for this angle, illuminating different color combinations to indicate whether an aircraft's approach is too high, too low, or just right. An ideal glide path for commercial airliners might be quite shallow, while more robust flying machines, like the space shuttle, require steep angles to safely and efficiently decelerate and land.