Problem 29 - College Physics

Question

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 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?

Step-by-Step Solution

Answer

(a) Yes, the time to reach Mach 3 is 20.26 s, longer than 5 s. (b) The greatest speed is 245 m/s.

Step 1: Understand the Problem

We are asked to determine if a pilot will black out when accelerating from rest at a rate of \(5g\) (where \(g\) is the acceleration due to gravity) to reach Mach 3 in 5 seconds. We are also to find out the maximum speed the pilot can attain within this acceleration period without blacking out.

Step 2: Convert Mach 3 to m/s

First, calculate the speed of Mach 3 using the speed of sound.\[\text{Mach 3 speed} = 3 \times 331 \text{ m/s} = 993 \text{ m/s}\]The pilot aims to reach a speed of 993 m/s.

Step 3: Calculate Acceleration in m/s²

Acceleration due to gravity is \(g = 9.8 \text{ m/s}^2\). Given the acceleration is \(5g\):\[a = 5 \times 9.8 = 49 \text{ m/s}^2\]

Step 4: Determine Time to Reach Mach 3

Use the formula \(v = u + at\) where \(v = 993 \text{ m/s}\), \(u = 0\), and \(a = 49 \text{ m/s}^2\):\[ 993 = 0 + 49t \]Solve for \(t\):\[ t = \frac{993}{49} \approx 20.26 \text{ s}\]This time exceeds 5 seconds, meaning he will black out before reaching Mach 3.

Step 5: Calculate Maximum Reachable Speed in 5 seconds

Use the formula \(v = u + at\) with \(u = 0\), \(a = 49 \text{ m/s}^2\), and \(t = 5 \text{ s}\):\[ v = 0 + 49 \times 5 = 245 \text{ m/s}\]

Key Concepts

Kinematics

Kinematics is the branch of physics that describes the motion of objects. It is mainly concerned with quantities like displacement, velocity, and acceleration, as well as their interrelationships. In kinematic equations, we assume constant acceleration to simplify the modeling and calculations of motion. The fundamental equations link these key aspects:

• Displacement ( \( s \)): The change in position of an object.
• Velocity ( \( v \)): The rate of change of position.
• Acceleration ( \( a \)): The rate of change of velocity.

These relationships can be represented using equations like: \[ v = u + at \] \[ s = ut + \frac{1}{2}at^2 \] where \( u \) is initial velocity, \( v \) is final velocity, \( t \) is time, and \( a \) is constant acceleration. By understanding these equations, one can predict the future position and velocity of any object under uniform acceleration, which is a core aspect of solving motion problems.

Motion Under Constant Acceleration

Motion under constant acceleration is a common scenario in physics, often used to analyze systems like free-falling objects, vehicles, or projectiles. Constant acceleration means that the acceleration value does not change as the body moves. This is greatly simplified by Newton's Second Law of Motion.
The acceleration due to gravity, denoted as \(g\), is a typical constant acceleration on Earth, measuring approximately \(9.8 \text{ m/s}^2\). In this exercise, the jet accelerates at \( 5g \), which implies: \[ a = 5 \times 9.8 = 49 \text{ m/s}^2 \] Using this constant acceleration, we can find the time to reach specific velocities or the maximum speed achievable within given time constraints.
These insights are vital in aviation, where calculating how quickly a pilot or jet can accelerate is crucial for their performance parameters and safety.

Jet Speed and G-Force

G-force is a measure of the force of gravity on an object, and in aviation, it represents the amount of force that pilots feel due to rapid acceleration or deceleration. The typical value of \(g\) is \(9.8 \text{ m/s}^2\), and when a pilot experiences \(5g\), the force felt is five times the normal gravitational pull.
Understanding this force is essential for pilots, as excessive g-forces can lead to blackouts or physical harm due to reduced blood flow. During extreme maneuvers, as in this problem where the jet speeds up from rest, pilots need to stay within safe g-force limits. Reaching the desired high speeds, such as Mach 3, must be balanced with endurance to maintain pilot alertness and safety.
In calculating these forces, the acceleration felt by the pilot is critical, and pilots must manage it wisely during flight to avoid the potential risks associated with prolonged high g-forces, ensuring safe and efficient operation.

Mach Number Conversion

The Mach number is a dimensionless unit used in fluid dynamics and aerodynamics to describe speeds in terms of the speed of sound. A Mach number tells us how many times the speed of sound something is traveling.
For example, Mach 1 equals the speed of sound, and in this scenario, Mach 3 indicates three times the speed of sound. Using the problem's provided speed of sound, \(331 \text{ m/s}\):

• Mach 1 = \(331 \text{ m/s}\)
• Mach 2 = \(662 \text{ m/s}\)
• Mach 3 = \(993 \text{ m/s}\)

This conversion helps aviators and engineers quantify how fast vehicles are traveling compared to the speed of sound, playing a critical role in design and safety. Being able to convert between Mach numbers and actual speeds allows for better planning and understanding of jet capabilities during varied flight conditions.