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Chapter 2: Problem 30

Blackout? A jet fighter pilot wishes to accelerate from rest at 5\(g\) to reach Mach 3 (three times the speed of sound) as quickly as possible. Experimental tests reveal that he will black out if this acceleration lasts for more than 5.0 s. Use 331 \(\mathrm{m} / \mathrm{s}\) for the speed of sound. (a) Will the period of acceleration last long enough to cause him to black out? (b) What is the greatest speed he can reach with an acceleration of 5\(g\) before blacking out?

Short Answer

Expert verified
(a) Yes, the pilot will black out; (b) Maximum speed is 245 m/s before blackout.

Step by step solution

01

Calculate Mach 3 Speed

The speed of sound is given as 331 m/s. To find Mach 3, multiply this speed by 3: \( v = 3 \times 331 = 993 \text{ m/s} \). This is the speed the pilot wants to reach.
02

Determine Acceleration in m/s虏

The problem states that the pilot accelerates at \( 5g \). Since gravity \( g \) is approximately \( 9.8 \text{ m/s}^2 \), the acceleration \( a \) is \( a = 5 \times 9.8 = 49 \text{ m/s}^2 \).
03

Calculate Time Required to Reach Mach 3

Using the formula for time with constant acceleration, \( t = \frac{v}{a} \), substitute the values: \( t = \frac{993}{49} \approx 20.3 \text{ seconds} \). This is the time required to reach Mach 3.
04

Determine If Blackout Will Occur

Since the time calculated to reach Mach 3 is \( 20.3 \text{ seconds} \), which is greater than the critical 5 seconds period, the pilot will definitely black out before reaching this speed.
05

Calculate Maximum Speed Before Blackout

Now, we find the maximum speed \( v \) the pilot can achieve in 5 seconds using \( v = a \cdot t \). Substitute \( a = 49 \text{ m/s}^2 \) and \( t = 5 \): \( v = 49 \times 5 = 245 \text{ m/s} \).
06

Compare Maximum Speed to Mach 3

The maximum speed before the pilot blacks out is 245 m/s, which is far less than Mach 3 (993 m/s).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Kinematics
Kinematics is a branch of classical mechanics focused on the motion of objects without considering the forces that cause this motion. In this problem, we deal with linear motion, where the jet fighter pilot accelerates from rest. Key kinematic equations help us relate acceleration, velocity, and time.
The major kinematic equation used here is:
  • \[ v = a \cdot t \]
  • \(v\): final velocity in meters per second (m/s).
  • \(a\): acceleration in meters per second squared (m/s虏).
  • \(t\): time in seconds (s).
By understanding this relationship, we can predict the pilot's speed at any moment during acceleration. Since kinematics focuses on such relationships, it is essential for calculating distances achieved under given acceleration conditions.
Acceleration
Acceleration is the rate at which an object's velocity changes with time. It is a vector quantity, which means it has both magnitude and direction. In this exercise, the acceleration is given as five times the gravitational acceleration (\(5g = 49 \text{ m/s}^2\)).
  • \(1g\) is equivalent to \(9.8 \text{ m/s}^2\).
  • The pilot accelerates at \(5 \times 9.8 = 49 \text{ m/s}^2\).
This high acceleration is necessary for reaching high speeds within a short period. However, rapid acceleration affects the pilot's body significantly, leading to physical limits such as blackouts if sustained beyond five seconds.
Mach Number
The Mach number is a dimensionless unit used in fluid mechanics to describe an object's speed in relation to the speed of sound. It is defined as:
  • \[ \text{Mach Number (M)} = \frac{\text{object speed}}{\text{speed of sound}} \]
In the exercise, Mach 3 means achieving a speed three times that of sound. The given speed of sound is 331 m/s. Thus, Mach 3 would be \(3 \times 331 \text{ m/s} = 993 \text{ m/s}\).
The Mach number is crucial for understanding how fast the pilot aims to go relative to acoustics of the air around him. High Mach numbers often reach supersonic speeds, where unique aerodynamic challenges arise.
Speed of Sound
The speed of sound is the speed at which sound waves travel through a medium. In this problem, the speed of sound is given as 331 m/s in air at standard conditions. It's a critical reference point for determining target speeds like Mach 3.
  • Depends on atmospheric conditions: temperature, pressure, and humidity.
  • Increases with higher temperatures because molecules move faster.
In physics problems, it provides a benchmark to measure supersonic or subsonic speeds. Here, using 331 m/s allows us to comprehensively calculate the speeds relevant to understand the pilot's acceleration and its limits when trying to reach Mach 3.

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Most popular questions from this chapter

(a) If a flea can jump straight up to a height of \(22.0 \mathrm{cm},\) what is its initial speed (in \(\mathrm{m} / \mathrm{s} )\) as it leaves the ground, neglecting air resistance? (b) How long is it in the air? (c) What are the magnitude and direction of its acceleration while it is (i) moving upward? (ii) moving downward? (iii) at the highest point?

On a 20 -mile bike ride, you ride the first 10 miles at an average speed of 8 \(\mathrm{mi} / \mathrm{h}\) . What must your average speed over the next 10 miles be to have your average speed for the total 20 miles be (a) 4 \(\mathrm{mi} / \mathrm{h} ?\) (b) 12 \(\mathrm{mi} / \mathrm{h} ?\) (c) Given this average speed for the first 10 miles, can you possibly attain an average speed of 16 \(\mathrm{mi} / \mathrm{h}\) for the total \(20-\) mile ride? Explain.

(a) The pilot of a jet fighter will black out at an acceleration greater than approximately 5\(g\) if it lasts for more than a few seconds. Express this acceleration in \(\mathrm{m} / \mathrm{s}^{2}\) and \(\mathrm{ft} / \mathrm{s}^{2}\) (b) The acceleration of the passenger during a car crash with an air bag is about 60\(g\) for a very short time. What is this acceleration in \(\mathrm{m} / \mathrm{s}^{2}\) and \(\mathrm{ft} / \mathrm{s}^{2}\) (c) The acceleration of a falling body on our moon is 1.67 \(\mathrm{m} / \mathrm{s}^{2} .\) How many \(g^{\prime}\) is this? (d) If the acceleration of a test plane is \(24.3 \mathrm{m} / \mathrm{s}^{2},\) how many \(g^{\prime}\) is it?

You and a friend start out at the same time on a 10 -km run. Your friend runs at a steady 2.5 \(\mathrm{m} / \mathrm{s} .\) How fast do you have to run if you want to finish the run 15 minutes before your friend?

Ouch! Nerve impulses travel at different speeds, depending on the type of fiber through which they move. The impulses for touch travel at \(76.2 \mathrm{m} / \mathrm{s},\) while those registering pain move at 0.610 \(\mathrm{m} / \mathrm{s}\) . If a person stubs his toe, find (a) the time for each type of impulse to reach his brain, and (b) the time delay between the pain and touch impulses. Assume that his brain is 1.85 m from his toe and that the impulses travel directly from toe to brain.
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