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Chapter 2: Problem 24

A pilot who accelerates at more than 4\(g\) begins to "gray out" but doesn't completely lose consciousness. (a) Assuming constant acceleration, 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 m/s for the speed of sound in cold air.)

Short Answer

Expert verified
(a) Shortest time is 33.8 seconds. (b) Distance is about 22395 meters.

Step by step solution

01

Define the problem and known values

We know the speed of sound in cold air is 331 m/s. The speed the pilot wants to achieve is Mach 4, which is four times the speed of sound: \( 4 \times 331 \text{ m/s} = 1324 \text{ m/s} \). The maximum acceptable acceleration is 4 times the gravitational acceleration, \( 4g = 4 \times 9.8 \text{ m/s}^2 = 39.2 \text{ m/s}^2 \). The initial velocity \( v_0 \) is 0 since the jet starts from rest.
02

Calculate the shortest time to reach Mach 4

Use the formula for acceleration: \( v = v_0 + at \). Here, \( v_0 = 0 \) and \( a = 39.2 \text{ m/s}^2 \). Set \( v = 1324 \text{ m/s} \) and solve for \( t \):\[ t = \frac{v}{a} = \frac{1324}{39.2} \approx 33.8 \text{ seconds} \]
03

Calculate the distance traveled during acceleration

Use the formula for distance: \( s = v_0 t + \frac{1}{2} a t^2 \). Since \( v_0 = 0 \):\[ s = \frac{1}{2} \times 39.2 \times (33.8)^2 \] Compute the distance:\[ s \approx \frac{1}{2} \times 39.2 \times 1142.44 \approx 22395 \text{ meters} \]

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Key Concepts

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

Acceleration
Acceleration is a fundamental concept in kinematics. It refers to the rate at which an object's velocity changes over time. In the context of a pilot accelerating in a jet, it's important to consider limits to prevent physical harm.

Key points about acceleration:
  • Measured in meters per second squared \( \text{m/s}^2 \).
  • Acceleration towards speeds that are a multiple of the gravitational constant, known as \(g\), must be monitored for safety.
  • In our scenario, the maximum allowable acceleration is \(4g\), which translates to \(39.2 \text{ m/s}^2\).
When accelerating from rest to a certain velocity, the time taken can be determined using the equation: \( t = \frac{v}{a} \), where \(v\) is the final velocity, and \(a\) is the acceleration. Understanding how acceleration impacts the time to reach specific velocities is crucial in kinematics.
Mach Number
The Mach number is a dimensionless quantity representing the ratio between the speed of an object and the speed of sound in the surrounding medium. It is a critical concept in aerodynamics.

Key aspects of Mach number:
  • Mach 1 is the speed of sound; thus, Mach 4 is four times the speed of sound.
  • In the exercise, the speed of sound is given as \(331 \text{ m/s}\), so Mach 4 equals \(1324 \text{ m/s}\).
  • Understanding Mach numbers is essential for computing velocities that reach or exceed supersonic speeds.
Mach numbers are widely used in aerospace engineering to classify aircraft speed, and achieving high Mach numbers requires significant acceleration.
Speed of Sound
The speed of sound is the velocity at which sound waves travel through a medium. In this exercise, we are dealing with sound speed in cold air.

Key information about the speed of sound:
  • The speed of sound in cold air is approximately \(331 \text{ m/s}\).
  • It varies with temperature and atmospheric conditions.
  • Understanding the speed of sound is essential for calculating Mach numbers and the corresponding velocities.
The speed of sound is central to aerodynamics and impacts calculations involving velocity, acceleration, and distance, especially at supersonic speeds.
Distance Traveled
Distance traveled is a core concept when analyzing motion under acceleration. It tells us how far an object moves over a period of time.

Steps to calculate distance during accelerated motion:
  • Use the formula for distance: \(s = v_0 t + \frac{1}{2} a t^2 \).
  • In our case, the initial velocity \(v_0\) is zero, simplifying the formula to \(s = \frac{1}{2} a t^2 \).
  • Plug in the acceleration \(a = 39.2 \text{ m/s}^2\) and time \(t \approx 33.8 \text{ s}\) to calculate.
  • The computed distance traveled is approximately \(22395 \text{ meters}\).
By understanding how to compute the distance from constant acceleration, one gains insights into how motion unfolds over time.

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Most popular questions from this chapter

If the contraction of the left ventricle lasts 250 ms and the speed of blood flow in the aorta (the large artery leaving the heart) is 0.80 m/s at the end of the contraction, what is the average acceleration of a red blood cell as it leaves the heart? (a) 310 ms\(^2\); (b) 31 m/s\(^2\); (c) 3.2 m/s\(^2\); (d) 0.32 m/s\(^2\).

A certain volcano on earth can eject rocks vertically to a maximum height \(H\). (a) How high (in terms of \(H\)) would these rocks go if a volcano on Mars ejected them with the same initial velocity? The acceleration due to gravity on Mars is 3.71 m/s\(^2\); ignore air resistance on both planets. (b) If the rocks are in the air for a time \(T\) on earth, for how long (in terms of \(T\)) would they be in the air on Mars?

You are climbing in the High Sierra when you suddenly find yourself at the edge of a fog-shrouded cliff. To find the height of this cliff, you drop a rock from the top; 8.00 s later you hear the sound of the rock hitting the ground at the foot of the cliff. (a) If you ignore air resistance, how high is the cliff if the speed of sound is 330 m/s? (b) Suppose you had ignored the time it takes the sound to reach you. In that case, would you have overestimated or underestimated the height of the cliff? Explain.

A 7500-kg rocket blasts off vertically from the launch pad with a constant upward acceleration of 2.25 m/s\(^2\) and feels no appreciable air resistance. When it has reached a height of 525 m, its engines suddenly fail; the only force acting on it is now gravity. (a) What is the maximum height this rocket will reach above the launch pad? (b) How much time will elapse after engine failure before the rocket comes crashing down to the launch pad, and how fast will it be moving just before it crashes? (c) Sketch \(a_y-t, v_y-t\), and \(y-t\) graphs of the rocket's motion from the instant of blast-off to the instant just before it strikes the launch pad.

A large boulder is ejected vertically upward from a volcano with an initial speed of 40.0 m/s. Ignore air resistance. (a) At what time after being ejected is the boulder moving at 20.0 m/s upward? (b) At what time is it moving at 20.0 m/s downward? (c) When is the displacement of the boulder from its initial position zero? (d) When is the velocity of the boulder zero? (e) What are the magnitude and direction of the acceleration while the boulder is (i) moving upward? (ii) Moving downward? (iii) At the highest point? (f) Sketch \(a_y-t, v_y-t\), and \(y-t\) graphs for the motion.
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