Problem 31 - College Physics

Question

If a pilot accelerates at more than \(4 g\), he begins to "gray out" but does not completely lose consciousness. (a) What is the shortest time that a jet pilot starting from rest can take to reach Mach 4 (four times the speed of sound) without graying out? (b) How far would the plane travel during this period of acceleration? (Use \(331 \mathrm{~m} / \mathrm{s}\) for the speed of sound.)

Step-by-Step Solution

Answer

(a) Approximately 33.73 seconds (b) Approximately 22,359.1 meters

Step 1: Understand Maximum Acceleration

The maximum acceleration the pilot can withstand without graying out is more than \(4g\), where \(g\) is the acceleration due to gravity \(9.81 \text{ m/s}^2\). Therefore, the maximum tolerable acceleration is \(4 \times 9.81 = 39.24 \text{ m/s}^2\).

Step 2: Calculate Final Velocity for Mach 4

Mach 4 is four times the speed of sound. Given that the speed of sound is \(331 \text{ m/s}\), the final velocity the pilot needs to reach is \(v = 4 \times 331 = 1324 \text{ m/s}\).

Step 3: Find Shortest Time Using Maximum Acceleration

Using the equation \(v = at\), where \(v = 1324 \text{ m/s}\) and \(a = 39.24 \text{ m/s}^2\), solve for \(t\): \[\frac{1324}{39.24} \approx 33.73 \text{ seconds}\]This is the shortest time the pilot can take to reach Mach 4 without graying out.

Step 4: Calculate the Distance Traveled During Acceleration

Using the equation for distance \(d = \frac{1}{2} at^2\), substitute \(a = 39.24 \text{ m/s}^2\) and \(t = 33.73 \text{ s}\): \[d = \frac{1}{2} \times 39.24 \times (33.73)^2\]\[d \approx 22359.1 \text{ meters}\]Thus, the plane travels approximately 22,359.1 meters during this acceleration period.

Key Concepts

Acceleration

Acceleration plays a crucial role in determining how quickly a jet can reach a particular speed. Here, it refers to the rate at which the pilot's speed increases while flying the jet. The maximum acceleration without "graying out," which means losing partial consciousness, is more than 4 times the acceleration due to gravity, denoted as \(4g\). The standard gravity \(g\) measures \(9.81 \, \text{m/s}^2\). Therefore, the maximum permissible acceleration is \(4 \times 9.81 = 39.24 \, \text{m/s}^2\). When calculating acceleration, it's essential to find the balance where the pilot can increase speed rapidly, yet safely. This value defines the upper limit of what the pilot can physically endure without risk.

Speed of Sound

The speed of sound is a critical benchmark in aerodynamics and is often used to measure high-speed flight. In the context of the problem, the speed of sound is given as \(331 \, \text{m/s}\). When an object travels faster than the speed of sound, it's said to be moving at "Mach" speeds.In this exercise, the goal is to reach Mach 4, which is four times the speed of sound. Therefore, the target velocity for the pilot is \(4 \times 331 = 1324 \, \text{m/s}\). Understanding Mach speeds is crucial for pilots and engineers when designing and flying high-speed aircraft, as it helps predict the aerodynamic forces and necessary structural integrity of the vehicle.

Distance Calculation

Once the rate of acceleration and the target velocity are determined, calculating the distance traveled during acceleration is straightforward. Using the formula \(d = \frac{1}{2}at^2\), where \(a\) represents acceleration and \(t\) is time, we can find the distance.In the problem, with an acceleration of \(39.24 \, \text{m/s}^2\) over \(33.73 \, \text{seconds}\), the distance becomes: \[ d = \frac{1}{2} \times 39.24 \times (33.73)^2 \]Calculating this gives a distance of approximately \(22359.1 \, \text{meters}\). Knowing this traveled distance helps in planning flight paths, ensuring the aircraft remains within safe and controlled operational limits.

G-Force

G-Force refers to the force acting on a body due to acceleration relative to free-fall or gravity. It is a crucial concept for pilots especially when dealing with high-speed jets.Here, the pilot experiences acceleration that simulates four times the gravitational force \(4g\), where \(1g\) equals \(9.81 \, \text{m/s}^2\). Pilots train to endure these forces while maintaining consciousness and control of their aircraft. Understanding G-Force is essential for safety, as excessive force can lead to a physiological response known as "graying out" or even blackouts. This aspect of physics ensures that aircraft operate within human limits, protecting pilots from harmful forces while enabling extraordinary flight capabilities.