Problem 56 - Physics for Scientists and Engineers with Modern Physics

Question

(II) A pilot performs an evasive maneuver by diving vertically at \(310 \mathrm{~m} / \mathrm{s}\). If he can withstand an acceleration of 9.0 's's without blacking out, at what altitude must he begin to pull out of the dive to avoid crashing into the sea?

Step-by-Step Solution

Answer

The pilot must begin to pull out of the dive at 1088 meters altitude.

Step 1: Understand the Problem

We need to determine the altitude at which the pilot must start pulling up from a vertical dive to withstand a maximum acceleration of 9.0 g's (where g is the acceleration due to gravity, approximately 9.81 m/s²).

Step 2: Identify and Set Up the Formula

The maximum acceleration when pulling out of a dive is given by the centripetal acceleration formula: \( a = \frac{v^2}{r} \), where \( a \) is the acceleration, \( v \) is the velocity, and \( r \) is the radius of the path. Here, \( a \) should not exceed \( 9 \times 9.81 \text{ m/s}^2 \).

Step 3: Substitute the Known Values

We are given that the speed \( v = 310 \text{ m/s} \), and \( a = 9 \times 9.81 = 88.29 \text{ m/s}^2 \). Therefore, substituting these into \( 88.29 = \frac{310^2}{r} \), we can solve for \( r \).

Step 4: Solve for the Radius of Curvature

Rearrange the equation to find \( r \): \[ r = \frac{v^2}{a} = \frac{310^2}{88.29} \].Calculate \( r \).

Step 5: Calculate the Minimum Altitude

Using \( v^2 = 96100 \text{ m}^2/\text{s}^2 \) and \( a = 88.29 \text{ m/s}^2 \), compute \( r \): \[ r = \frac{96100}{88.29} \approx 1088 \text{ m} \]. This means the pilot must start pulling out of the dive at a minimum altitude of 1088 meters.

Key Concepts

Evasive Maneuver

When a pilot is performing an evasive maneuver, they are executing a rapid movement or change in flight path to avoid a potential threat or obstacle. In this situation, the pilot needs to react quickly to maintain their trajectory and avoid a crash. Evasive maneuvers often involve sharp turns, dives, or climbs, requiring precision and coordination.
In this exercise, the pilot dives vertically, a common technique to gain speed quickly. However, pulling out of such a dive necessitates careful calculation to prevent excessive forces on the body, which could lead to blacking out.
Understanding these maneuvers is crucial for pilots, as they must balance rapid actions with the physiological limits of both the aircraft and themselves. Proper execution involves understanding both the aircraft's capabilities and the pilot's own physical limitations.

Acceleration Limit

The human body can only withstand a certain level of acceleration before it becomes dangerous. In aviation, pilots are trained to manage these limits, often referred to in terms of "g-forces." A g-force is a measure of acceleration relative to the force of gravity on Earth.
For this exercise, the pilot can endure an acceleration of up to 9 g's. Since 1 g equates to approximately 9.81 m/s² (gravitational acceleration on Earth), the maximum tolerable acceleration is 88.29 m/s².
Exceeding this limit can lead to a blackout as blood flow to the brain becomes compromised. This makes understanding and calculating acceleration limits critical, especially during high-stakes maneuvers. Pilots must continually assess their acceleration to ensure it remains within safe limits.

Altitude Calculation

In this exercise, determining the correct altitude for initiating a pull-up out of the dive is essential to avoid hitting the ground. The calculation relies on the concept of centripetal acceleration, which is the rate of change of tangential velocity for an object moving in a circular path. The formula used is \( a = \frac{v^2}{r} \), where \( v \) is velocity and \( r \) is the radius of curvature, representing the path's curvature length in a dive.
By dividing the square of the velocity \( (310^2) \) by the allowed acceleration limit \( (88.29 \text{ m/s}^2) \), the result is the radius \( r \): approximately 1088 meters. This value represents the necessary distance from the ground where the pilot should begin to modify their flight path to safely transition from a dive.
Calculating the correct altitude is crucial for ensuring the maneuver can be completed safely without exceeding the aircraft's acceleration limits.