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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 \(\mathrm{m} / \mathrm{s}\) 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) The period of acceleration is not enough to reach Mach 3 before blacking out. (b) Maximum speed without blacking out: 245.25 m/s.

Step by step solution

01

Understanding Given Values and Constants

The pilot accelerates from rest, so the initial velocity \( v_0 = 0 \). The acceleration \( a = 5g \), where \( g = 9.81 \ \text{m/s}^2 \), resulting in \( a = 5 \times 9.81 = 49.05 \ \text{m/s}^2 \). The maximum acceptable duration is 5 s. The speed of sound is 331 m/s, meaning Mach 3 is \( 3 \times 331 = 993 \ \text{m/s} \).
02

Calculate Maximum Velocity in Given Time

Using the kinematic equation for velocity \( v = v_0 + at \), where \( v_0 = 0 \), \( a = 49.05 \ \text{m/s}^2 \), and \( t = 5 \ \text{s} \), calculate the greatest speed:\[ v = 0 + 49.05 \times 5 = 245.25 \ \text{m/s} \]
03

Compare Maximum Velocity with Mach 3

The calculated velocity is 245.25 m/s. Compare this with the velocity for Mach 3 (993 m/s) to check if it can be reached in 5 seconds. Since 245.25 m/s < 993 m/s, Mach 3 cannot be achieved within the 5-second limit.
04

Conclusion From Calculations

Since the pilot can only reach 245.25 m/s in 5 seconds and this is less than Mach 3, the acceleration period does not allow reaching Mach 3 without exceeding the time limit. Thus, the pilot will not black out due to reaching Mach 3.
05

Revisiting the Blackout Condition

The condition for the pilot to blackout applies for accelerations lasting more than 5 seconds, but the greatest speed the pilot can reach without blacking out is 245.25 m/s (less than Mach 3). Thus, there is no risk of blacking out within this set period.

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Key Concepts

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

Jet Fighter Pilot Acceleration
When a jet fighter pilot flies a high-speed aircraft, the forces they experience are immense. Acceleration is a key aspect while examining these forces, particularly when a pilot needs to accelerate quickly from rest. In this exercise, the pilot experiences a constant acceleration of 5 times the acceleration due to gravity, denoted as 5g. Here's what you need to know:
  • The typical gravitational acceleration, denoted as g, is about 9.81 m/s虏.
  • Applying 5g means the pilot experiences an acceleration of 49.05 m/s虏 (calculated as 5 times 9.81 m/s虏).
  • Constant acceleration over time can rapidly change a jet fighter's speed, but it must be balanced with the pilot's physical limits to avoid health risks like blackouts.
In this problem, the pilot can withstand this intense acceleration for a maximum of 5 seconds before risking a blackout.
Mach Number
The Mach number is a dimensionless unit used to compare an object's speed with the speed of sound. It's vital for understanding how fast an aircraft is moving relative to the sound barrier. Here's how it works:
  • Mach 1 is the speed of sound, which is around 331 m/s in standard conditions.
  • An object moving at Mach 3 would be traveling at three times the speed of sound, or about 993 m/s in this problem (3 times 331 m/s).
  • Mach numbers help determine the capabilities of aircraft in terms of speed, especially those exceeding the speed of sound, known as supersonic speeds.
In this exercise, the goal was to reach Mach 3, but time constraints prevented achieving this speed.
Speed of Sound
Understanding the speed of sound is crucial in aerodynamics and kinematics, especially for high-speed jets.
  • The speed of sound is the rate at which sound waves travel through the air, typically 331 m/s at sea level and at 0掳C.
  • This speed can vary with atmospheric conditions such as altitude and temperature.
  • Achieving speeds beyond the speed of sound requires overcoming additional air resistance and sonic barriers.
In this problem, the speed of sound is used as a benchmark to determine Mach numbers and evaluate the aircraft's maximum attainable speed.
Kinematic Equations
Kinematic equations are fundamental in physics for predicting the motion of objects under uniform acceleration. These equations establish relationships between velocity, acceleration, time, and displacement. Relevant equations include:
  • Velocity equation: \( v = v_0 + at \)
  • Displacement equation: \( s = v_0t + \frac{1}{2}at^2 \)
  • Velocity squared equation: \( v^2 = v_0^2 + 2as \)
In this exercise, we used the velocity equation to calculate the maximum speed the pilot could reach in 5 seconds. Starting from rest (initial velocity zero), applying a constant acceleration, and calculating time helped determine the pilot's final velocity.
Acceleration Due to Gravity
Gravitational acceleration, usually denoted as g, plays a critical role in determining forces acting on a pilot during acceleration.
  • On Earth, the acceleration due to gravity is approximately 9.81 m/s虏.
  • This is a standard measure of how fast objects accelerate towards the Earth's surface due to gravity alone.
  • In aircraft, calculations often use g as a unit to express other forces or accelerations, like in this exercise where the pilot experienced 5g.
Understanding g is essential for evaluating the physical effects on pilots and the aircraft, essential for safety and performance assessments.

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Most popular questions from this chapter

A race car starts from rest and travels east along a straight and level track. For the first 5.0 s of the car's motion, the eastward component of the car's velocity is given by \(v_{x}(t)=\left(0.860 \mathrm{m} / \mathrm{s}^{3}\right) t^{2} .\) What is the acceleration of the car when \(v_{x}=16.0 \mathrm{m} / \mathrm{s} ?\)

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

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.

Dan gets on Interstate Highway \(\mathrm{I}-80\) at Seward, Nebraska, and drives due west in a straight line and at an average velocity of magnitude 88 \(\mathrm{km} / \mathrm{h}\) . After traveling 76 \(\mathrm{km}\) , he reaches the Aurora exit (Fig. \(\mathrm{P} 2.63 ) .\) Realizing he has gone too far, he turns around and drives due east 34 \(\mathrm{km}\) back to the York exit at an average velocity of magnitude 72 \(\mathrm{km} / \mathrm{h}\) . For his whole trip from Seward to the York exit, what are (a) his average speed and (b) the magnitude of his average velocity?

Passing. The driver of a car wishes to pass a truck that is traveling at a constant speed of 20.0 \(\mathrm{m} / \mathrm{s}\) (about 45 \(\mathrm{mi} / \mathrm{h} )\) . Initially, the car is also traveling at 20.0 \(\mathrm{m} / \mathrm{s}\) and its front bumper is 24.0 \(\mathrm{m}\) behind the truck's rear bumper. The car accelerates at a constant 0.600 \(\mathrm{m} / \mathrm{s}^{2}\) , then pulls back into the truck's lane when the rear of the car is 26.0 \(\mathrm{m}\) ahead of the front of the truck. The car is 4.5 \(\mathrm{m}\) long and the truck is 21.0 \(\mathrm{m}\) long. (a) How much time is required for the car to pass the truck? (b) What distance does the car travel during this time? (c) What is the final speed of the car?
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