Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

A catapult launcher on aircraft carrier accelerates a jet from rest to \(72 \mathrm{~m} / \mathrm{s}\). The work done by the catapult during the launch is \(7.6 \times 10^{7} \mathrm{~J}\). (a) What is the mass of the jet? (b) If the jet is in contact with the catapult for \(2.0 \mathrm{~s}\), what is the power output of the catapult?

Short Answer

Expert verified
Mass of the jet is 2939.5 kg and the power output is \(3.8 \times 10^7\) W.

Step by step solution

01

Understanding Work and Kinetic Energy

The work-energy principle states that the work done on an object is equal to the change in its kinetic energy. The equation for kinetic energy, \( KE \), is \( KE = \frac{1}{2}mv^2 \), where \( m \) is the mass and \( v \) is the velocity.
02

Calculate Mass of Jet

Since all the work done by the catapult is converted into the kinetic energy of the jet, we have \( W = \Delta KE = \frac{1}{2}mv^2 \). Given \( W = 7.6 \times 10^7 \) J and \( v = 72 \) m/s, we solve for \( m \) as follows: \[ m = \frac{2W}{v^2} = \frac{2 \times 7.6 \times 10^7}{72^2} = 2939.5 \text{ kg} \]
03

Calculate Power Output of Catapult

Power is the rate at which work is done, defined as \( P = \frac{W}{t} \), where \( t \) is the time. Given \( W = 7.6 \times 10^7 \) J and \( t = 2.0 \) s, the power output \( P \) is calculated as: \[ P = \frac{7.6 \times 10^7}{2.0} = 3.8 \times 10^7 \text{ W} \]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with Vaia!

Key Concepts

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

Kinetic Energy
Kinetic energy is a key aspect of physics that describes the energy possessed by an object in motion. Simply put, it's the energy an object has because of its velocity. The formula to calculate kinetic energy is given by:\[ KE = \frac{1}{2}mv^2 \]where:
  • \( KE \) represents the kinetic energy, measured in joules (J).
  • \( m \) is the mass of the object in kilograms (kg).
  • \( v \) stands for the velocity of the object in meters per second (m/s).

This principle is crucial in understanding scenarios like how a catapult launcher on an aircraft carrier propels a jet. The work done on the jet by the catapult is used entirely to increase the jet's kinetic energy. Thus, all this work translates into the jet achieving high speed from rest. Hence, kinetic energy plays a vital role in comprehending motion and energy transformation.
Power Output
Power output is the measure of the rate at which work is done. It demonstrates how quickly or slowly the energy conversion process occurs. In this context, power is calculated using the following formula:\[ P = \frac{W}{t} \]where:
  • \( P \) is the power in watts (W).
  • \( W \) represents the work done, also in joules (J).
  • \( t \) is the time in seconds (s) over which this work is performed.

In the scenario of the jet on an aircraft carrier, knowing how long the catapult is in contact with the jet helps determine the power output. The given problem shows us that in just 2 seconds, the catapult does significant work, reflecting a high power output of \( 3.8 \times 10^7 \) watts.This high rate of energy transfer allows the jet to reach its required speed swiftly. Understanding power helps us appreciate the effectiveness and efficiency of different mechanical processes and technologies.
Mass Calculation
Determining mass using work-energy principles involves understanding how the work performed impacts the object's state of motion. The relationship derives from rearranging the equation for kinetic energy:\[ W = \frac{1}{2}mv^2 \]Given it equals the work done on an object, rearranging for mass \( m \) leads to:\[ m = \frac{2W}{v^2} \]Here:
  • \( W \) is the work done, given as \( 7.6 \times 10^7 \) joules.
  • \( v \) is the jet's final velocity at \( 72 \) m/s.

By plugging in these values, the mass of the jet is calculated as approximately \( 2939.5 \) kg. This formula proves that by knowing the work done and the achieved speed, one can deduce the object's mass.This concept is vital for scenarios where direct measurement of mass is impractical or impossible, allowing us to infer mass from motion and energy data.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

See all solutions

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free