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Chapter 2: Problem 20

A jet fighter pilot wishes to accelerate from rest at a constant acceleration of 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 m/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 he blacks out?

Short Answer

Expert verified
(a) Yes, blackout will occur; it takes 20.3 s to reach Mach 3. (b) Greatest speed is 245 m/s.

Step by step solution

01

Define Constants and Known Values

First, we define the constants given in the problem. The acceleration due to gravity, denoted by \(g\), is approximately 9.8 m/s虏. Therefore, an acceleration of 5\(g\) is equal to \(5 \times 9.8 = 49\) m/s虏. The speed of sound is given as 331 m/s, so Mach 3 is \(3 \times 331 = 993\) m/s.
02

Calculate Time Required to Reach Mach 3

We need to calculate the time required to reach Mach 3 from rest with a constant acceleration of 49 m/s虏. Using the formula for acceleration, \(v = u + at\), where \(v\) is the final velocity, \(u\) is the initial velocity, \(a\) is acceleration, and \(t\) is time, set \(u = 0\) (since starting from rest), \(v = 993\) m/s, and \(a = 49\) m/s虏.Rearranging the equation for \(t\):\[t = \frac{v}{a} = \frac{993}{49} \approx 20.3 \text{ s}.\]
03

Assess Blackout Risk

The problem states that the pilot will black out if the acceleration lasts more than 5.0 s. From Step 2, we found it would take approximately 20.3 s to reach Mach 3 at this acceleration, which is much longer than 5.0 s. Hence, the pilot will indeed black out before reaching Mach 3.
04

Calculate Greatest Speed Without Blackout

Next, calculate the greatest speed the pilot can achieve within 5.0 s at an acceleration of 49 m/s虏. Using the formula \(v = u + at\) again with \(t = 5\) s:\[v = 0 + 49 \times 5 = 245 \text{ m/s}.\]
05

Conclusion

(a) The acceleration period will last long enough to cause blackout, as it takes about 20.3 s to reach Mach 3, but the pilot can only withstand it for 5.0 s. (b) The greatest speed before blacking out is 245 m/s.

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Key Concepts

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

Acceleration
Acceleration is the rate at which an object's velocity changes over time. In simple terms, it's how fast something speeds up or slows down. Acceleration is a vector, meaning it has both a magnitude and a direction.
In this exercise, a fighter jet accelerates from a standstill with a constant acceleration of 5 times the gravitational force (5\(g\)). The unit \(g\) stands for the acceleration due to Earth's gravity and has a value of approximately 9.8 m/s虏. Hence, 5\(g\) equals \(49 \text{ m/s}^2\).
  • Positive acceleration means an increase in velocity.
  • Negative acceleration (deceleration) means a decrease in velocity.
  • Zero acceleration means constant velocity.
Understanding acceleration helps us calculate how fast the jet can reach a given speed or "Mach number."
Mach Number
The Mach number is a dimensionless unit that represents the ratio of an object's speed to the speed of sound in the surrounding medium. Mach numbers are commonly used when discussing high-speed travel such as supersonic flight, where speeds exceed the speed of sound.
The formula for Mach number is:
\[\text{Mach number} = \frac{\text{speed of object}}{\text{speed of sound}}\]
  • A Mach number less than one is subsonic (slower than sound).
  • A Mach number of one is exactly the speed of sound, termed as "sonic."
  • Above Mach 1 is supersonic, while speeds reaching or exceeding Mach 5 are hypersonic.
In this exercise, the pilot's goal is to reach Mach 3, which is three times the speed of sound, calculated as \(3 \times 331 = 993 \text{ m/s}\). Reaching such speeds in fighter jets is a testament to their advanced engineering capabilities.
Speed of Sound
The speed of sound is the speed at which sound waves travel through a medium. This varies based on factors such as medium type and temperature. In air at sea level, and under typical atmospheric conditions, the speed of sound is approximately 331 meters per second.
  • In denser media, like water and solids, the speed of sound is generally higher than in air.
  • Temperature increases result in higher speeds of sound in air since molecules move more quickly.
Sound speed plays a vital role in various applications, ranging from aviation to environmental studies. In our exercise, it's essential for understanding how fast the jet pilot needs to travel to reach Mach 3.
Kinematics
Kinematics is the branch of mechanics concerning the motion of objects without considering the forces that cause that motion. It helps us understand and predict how an object moves by using equations and principles.
The core equation used in this exercise is part of the basic kinematics equations:
\[v = u + at\]
Where:
  • \(v\) is the final velocity.
  • \(u\) is the initial velocity.
  • \(a\) is the acceleration.
  • \(t\) is the time.
Using this, we calculated both the time needed to reach Mach 3 and the greatest speed achievable under safe conditions. To ensure a consistent understanding, remember that kinematic equations assume uniform acceleration.

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Most popular questions from this chapter

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.

The human body can survive an acceleration trauma incident (sudden stop) if the magnitude of the acceleration is less than 250 m/s\(^{2}\). If you are in an automobile accident with an initial speed of 105 km/h (65 mi/h) and are stopped by an airbag that inflates from the dashboard, over what distance must the airbag stop you for you to survive the crash?

A subway train starts from rest at a station and accelerates at a rate of 1.60 m/s\(^2\) for 14.0 s. It runs at constant speed for 70.0 s and slows down at a rate of 3.50 m/s\(^2\) until it stops at the next station. Find the total distance covered.

A car sits on an entrance ramp to a freeway, waiting for a break in the traffic. Then the driver accelerates with constant acceleration along the ramp and onto the freeway. The car starts from rest, moves in a straight line, and has a speed of 20 m/s (45 mi/h) when it reaches the end of the 120-m-long ramp. (a) What is the acceleration of the car? (b) How much time does it take the car to travel the length of the ramp? (c) The traffic on the freeway is moving at a constant speed of 20 m/s. What distance does the traffic travel while the car is moving the length of the ramp?

A ball is thrown straight up from the ground with speed \(v_0\). At the same instant, a second ball is dropped from rest from a height \(H\), directly above the point where the first ball was thrown upward. There is no air resistance. (a) Find the time at which the two balls collide. (b) Find the value of \(H\) in terms of \(v_0\) and \(g\) such that at the instant when the balls collide, the first ball is at the highest point of its motion.
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