Problem 38 - College Physics

Question

\(\bullet\) Pilot blackout in a power dive. A jet plane comes in for a downward dive as shown in Figure \(3.39 .\) The bottom part of the path is a quarter circle having a radius of curvature of 350 \(\mathrm{m} .\) According to medical tests, pilots lose consciousness at an acceleration of 5.5\(g .\) At what speed (in \(\mathrm{m} / \mathrm{s}\) and mph) will the pilot black out for this dive?

Step-by-Step Solution

Answer

The pilot blacks out at about 137.37 m/s or 307.3 mph.

Step 1: Identify the Problem

We need to find the speed of the jet at which the pilot will black out due to experiencing a centripetal acceleration of 5.5 times the acceleration due to gravity, g. The path follows a quarter circle with a radius of 350 meters.

Step 2: Understand Acceleration Terms

Acceleration due to gravity, \( g \), is approximately \( 9.8 \, \text{m/s}^2 \). Thus, the acceleration causing blackout is \( 5.5g = 5.5 \times 9.8 \, \text{m/s}^2 \).

Step 3: Apply Centripetal Acceleration Formula

The formula for centripetal acceleration \( a_c \) is \( a_c = \frac{v^2}{r} \). We need to find the speed \( v \) such that the acceleration \( a_c = 5.5g \).

Step 4: Set Up Equation and Solve for Speed

Using the formula from the previous step:\[ \frac{v^2}{350} = 5.5 \times 9.8 \] Solve for \( v^2 \):\[ v^2 = 5.5 \times 9.8 \times 350 \]

Step 5: Calculate Speed in m/s

Solve\[ v = \sqrt{5.5 \times 9.8 \times 350} \] Calculate\[ v \approx \sqrt{18865} \approx 137.37 \, \text{m/s} \]

Step 6: Convert Speed to mph

There are approximately 2.237 miles in one meter per second. To convert speed from \( \text{m/s} \) to mph: \[137.37 \, \text{m/s} \times 2.237 \approx 307.3 \, \text{mph} \]

Step 7: Conclusion: Present the Result

The speed at which the pilot will black out is approximately 137.37 m/s or 307.3 mph.

Key Concepts

Pilot Blackout

Understanding why pilots experience blackout is crucial for aviation safety. When an aircraft maneuvers sharply, like in a power dive, the forces acting on the pilot increase dramatically. A pilot blackout occurs when these forces exceed the body's ability to circulate blood effectively, particularly to the brain. The body experiences "g" forces during such maneuvers, which is a measure of acceleration compared to the usual force of gravity we feel daily. Typically, humans can handle up to 5.5 times the normal gravitational force before blacking out. This is known as "g-force tolerance." If the pilot undergoes acceleration greater than this threshold, the pilot is at risk of losing consciousness.
For the given problem, if the centripetal acceleration - caused by the speed of the jet and the tightness of the curve it flies - exceeds 5.5 "g," the pilot can black out. Therefore, understanding this limit is critical for designing safe combat maneuvers for pilots.

Radius of Curvature

In aerodynamics, the radius of curvature is a crucial concept. It refers to the radius of the circular part of a path that an object follows, such as in a plane's dive. It basically describes how sharp or gentle a curve is.
The larger the radius, the more gradual the path, and vice versa. For the exercise, the dive follows a quarter circle with a radius of 350 meters. This means that the path taken by the plane is a gentle arc with a consistent radius throughout the maneuver.

• Understanding this concept is especially important for maneuvers that require high-speed curves because they affect centripetal acceleration.
• A smaller radius of curvature would result in a sharper turn and higher acceleration forces experienced by the plane and pilot.

The balance between speed, acceleration, and radius of curvature affects how safely and effectively aircraft can execute maneuvers.

Speed Conversion

When discussing speed, it's essential to recognize that different measurement units may be used depending on context or region. In the aviation field, speeds are often converted between meters per second (m/s) and miles per hour (mph).
In the problem, the speed at which blackout occurs is initially calculated in meters per second. However, for practical and comparative reasons, converting to miles per hour can also be insightful.

• The conversion factor used for this purpose is approximately 2.237. This means each meter per second is about 2.237 miles per hour.
• The formula involves multiplying the speed in m/s by this conversion factor to obtain the speed in mph.

For example, if the pilot blacks out at a speed of 137.37 m/s, converting this to mph gives approximately 307.3 mph. This conversion assists in assessing and comparing aircraft performance and capabilities in international scenarios.

Quarter Circle Path

The quarter circle path is a specific geometric maneuver in aviation where the aircraft follows the arc of a circle. It's essentially a 90-degree turn along the perimeter of a circle.
For the exercise, the plane's path through the lower portion of its dive is designed as a quarter circle with a specified radius of curvature. This maneuver results in predictable centripetal forces that can be crucial for controlling advanced maneuvers.

• In such paths, centripetal acceleration is directed towards the center of the circular path and is dependent on both the speed of the aircraft and the radius of the curve.
• Consistency in the quarter circle path allows for precise calculations of required speeds and forces, ensuring safety is maintained for pilots.

Understanding how these factors interplay helps in the design and execution of aerial maneuvers in combat or aerobatic flying, as well as in routine flight operations.