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Chapter 3: Problem 335

An electromagnetic catapult system is being designed to replace a steam-driven system on an aircraft carrier. The requirements include accelerating a 12000 -kg aircraft from rest to a speed of \(70 \mathrm{m} / \mathrm{s}\) over a distance of \(90 \mathrm{m} .\) What constant force \(F\) must the catapult exert on the aircraft?

Short Answer

Expert verified
The catapult must exert a force of approximately 326,640 N.

Step by step solution

01

Identify Known Values

The mass of the aircraft, \( m \), is 12000 kg. The final velocity, \( v_f \), is 70 m/s and the initial velocity, \( v_i \), is 0 m/s. The distance over which the acceleration occurs, \( d \), is 90 m.
02

Use Kinematic Equation to Find Acceleration

We use the kinematic equation \( v_f^2 = v_i^2 + 2a d \) to find the acceleration \( a \). Substitute the known values: \( 70^2 = 0 + 2a \times 90 \). Simplify and solve for \( a \): \( 4900 = 180a \), thus \( a = \frac{4900}{180} \approx 27.22 \ \mathrm{m/s^2} \).
03

Use Newton's Second Law to Find Force

Apply Newton's second law, \( F = ma \), to find the force. Substitute the mass \( m = 12000 \ \mathrm{kg} \) and the acceleration \( a = 27.22 \ \mathrm{m/s^2} \). Therefore, \( F = 12000 \times 27.22 \approx 326640 \head \mathrm{N}\).
04

Check Units and Reasonability

Verify that the units of force are in newtons (N). Since \( F = ma \) gives us kg路m/s虏, this is correct. Also, review the values: accelerating a 12000 kg aircraft to 70 m/s over 90 m with a force of approximately 326640 N is reasonable given the scenario of an aircraft carrier.

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Key Concepts

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

Newton's Second Law
Newton's Second Law is a fundamental principle in physics that connects force, mass, and acceleration. Simply put, it states that the force acting on an object is equal to the mass of the object multiplied by its acceleration, represented by the formula:
  • \( F = ma \)
This means that if you know the mass of an object and its acceleration, you can calculate the force applied to it.
For example, if you're trying to accelerate a large object, like an aircraft, you need a significant amount of force. This concept is crucial in understanding how different variables interact with each other to determine motion.
In practical terms, Newton's Second Law helps engineers and scientists predict the movement and behavior of objects under various forces. By rearranging the equation, you can solve for any of the three variables if the other two are known. This makes it a versatile tool for solving real-world problems.
Acceleration
Acceleration is the rate at which the velocity of an object changes over time. It is a vector quantity, meaning it has both magnitude and direction. In the context of the catapult system in our exercise, the goal is to accelerate an aircraft from rest to a designated speed over a specific distance.
To find acceleration, we used a kinematic equation which is derived from the basic principles of motion and includes other known values, such as distance and initial and final velocities:
  • \( v_f^2 = v_i^2 + 2ad \)
This formula incorporates the need to factor in the change in speed (from 0 to 70 m/s) and the distance of 90 meters over which this change occurs.
By solving this equation for \( a \), we found the acceleration to be approximately 27.22 m/s虏. Acceleration is crucial in this scenario because it determines the force necessary for the aircraft to reach the desired speed, influencing both engineering design and operational planning.
Force Calculation
Force Calculation involves determining the amount of force needed to achieve a certain level of acceleration for an object with a given mass. In the exercise, we used Newton's Second Law:
  • \( F = ma \)
This formula was instrumental in calculating the necessary force for the aircraft.
We substituted the mass of 12000 kg and the calculated acceleration of 27.22 m/s虏 into the formula to find that a force of approximately 326640 N (Newtons) is required. This step is crucial in design processes, such as determining the specifications for systems like the electromagnetic catapult.
Correct force calculation ensures that the machinery will perform as intended, achieving the necessary speed over the set distance without exceeding safety limits or causing mechanical failure. Understanding these calculations also allows for the optimization of fuel efficiency and resource use in engineering applications.

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Most popular questions from this chapter

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