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Catapult Launcher A catapult launcher on an aircraft carrier accelerates a jet from rest to 72 m>s. The work done by the catapult during the launch is 7.6 * 107 J. (a) What is the mass of the jet? (b) If the jet is in contact with the catapult for 2.0 s, what is the power output of the catapult?

Short Answer

Expert verified
(a) Mass of jet is approximately 29,323 kg. (b) Power output is 3.8 x 10^7 W.

Step by step solution

01

Understanding the Problem

We have been given the final speed of the jet after launch, the work done by the catapult, and the time the jet was in contact with the catapult. We need to find the jet's mass and the catapult's power output.
02

Identifying the Relevant Formula for Mass

To find the mass of the jet, we can use the work-energy principle, which states that the work done is equal to the change in kinetic energy. Thus, the formula is: \( W = \frac{1}{2} m v^2 \) where \( W = 7.6 \times 10^7 \text{ J} \) and \( v = 72 \text{ m/s} \). We need to solve for \( m \).
03

Solving for Mass of the Jet

Rearrange the formula: \( m = \frac{2W}{v^2} \). Substitute \( W = 7.6 \times 10^7 \text{ J} \) and \( v = 72 \text{ m/s} \) into the equation: \( m = \frac{2 \times 7.6 \times 10^7}{72^2} \). Calculate to find \( m \).
04

Calculating Mass

Calculate \( 72^2 = 5184 \). Then, \( m = \frac{2 \times 7.6 \times 10^7}{5184} = \frac{1.52 \times 10^8}{5184} \approx 29,323.4 \text{ kg} \). Thus, the mass of the jet is approximately 29,323 kg.
05

Formula for Power Output

Power is defined as the work done per unit time. Therefore, use the formula: \( P = \frac{W}{t} \). Here, \( W = 7.6 \times 10^7 \text{ J} \) and \( t = 2.0 \text{ s} \).
06

Calculating Power Output

Substitute the values into the formula: \( P = \frac{7.6 \times 10^7}{2} = 3.8 \times 10^7 \text{ W} \). Thus, the power output of the catapult is \( 3.8 \times 10^7 \text{ watts} \).

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Key Concepts

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

Kinetic Energy
Kinetic energy is a form of energy associated with the motion of an object. It depends on two primary factors:
  • the mass of the object
  • the speed of the object
When an object is set into motion, work is done to change its kinetic energy. This is essential in systems like the catapult launcher on an aircraft carrier, which accelerates a jet from rest. The formula for kinetic energy is given by: \[ KE = \frac{1}{2} mv^2 \]where:
  • \( KE \) is the kinetic energy
  • \( m \) is the mass of the object
  • \( v \) is the velocity of the object
In the exercise, calculating the jet's kinetic energy involves considering the work done by the catapult. This work results in the energy that gives the jet its final speed.
Power Output
Power output refers to the rate at which energy is transferred or converted. It's a measure of how fast work is done. When discussing mechanical processes like the catapult launching a jet, power is crucial because it tells us how quickly the energy is being used or delivered. The formula to compute power is:\[ P = \frac{W}{t} \]where:
  • \( P \) is the power output
  • \( W \) is the work done
  • \( t \) is the time period over which the work is done
Under the conditions given, the work done by the catapult over 2 seconds allows us to determine the power. Knowing the power output helps optimize the design for energy efficiency and performance during launch.
Mass Calculation
Calculating the mass of an object when given kinetic energy requires rearranging the formula that connects work and kinetic energy, as per the work-energy principle. In this problem, we're using:\[ W = \frac{1}{2} mv^2 \]Rearranging for mass \( m \) , we get:\[ m = \frac{2W}{v^2} \]This is useful when you know the work done and the final velocity. With this equation, we can solve for the jet's mass after launch.For instance, knowing that the catapult does work \( 7.6 \times 10^7 \) joules, and reaches a velocity of 72 m/s, substitution into the formula provides:\[ m = \frac{2 \times 7.6 \times 10^7}{72^2} \approx 29,323 \text{ kg} \]Calculating the mass in this manner was crucial so the calculations matched with the remaining physical setup and parameters given in the problem.

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Most popular questions from this chapter

(a) At what rate must you lift a 3.6-kg container of milk (1 gallon) if the power output of your arm is to be 22 W? (b) How long does it take to lift the milk container through a distance of 1.0 m at this rate?

Assess System 1 has a force of \(10 \mathrm{~N}\) and a speed of \(5 \mathrm{~m} / \mathrm{s}\). System 2 has a force of \(20 \mathrm{~N}\) and a speed of \(2 \mathrm{~m} / \mathrm{s}\). Which system has the greater power? Explain.

Human-Powered Flight Human-powered aircraft require a pilot to pedal, as on a bicycle, and to produce a sustained power output of about \(0.30 \mathrm{hp}(1 \mathrm{hp}=746 \mathrm{~W})\). The Gossamer Albatross flew across the English Channel on June 12,1979 , in \(2 \mathrm{~h} 49 \mathrm{~min}\). (a) How much energy did the pilot expend during the flight? (b) How many candy bars (280 Cal per bar) would the pilot have to consume to be "fueled up" for the flight? Note that a nutritional calorie (1 Cal) is equivalent to 1000 calories (1000 cal) as defined in physics. In addition, the conversion factor between calories and joules is as follows: \(1 \mathrm{Cal}=1000 \mathrm{cal}=1 \mathrm{kcal}=4186 \mathrm{~J} .\)

A block with a mass of \(3.7 \mathrm{~kg}\) slides with a speed of \(2.2 \mathrm{~m} / \mathrm{s}\) on a frictionless surface. The block runs into a stationary spring and compresses it a certain distance before coming to rest. What is the compression distance, given that the spring has a spring constant of \(3200 \mathrm{~N} / \mathrm{m}\) ?

Research and write a report on the power that humans produce in everyday life. Include the power produced by the brain when thinking and the heart when resting. Also select several strenuous activities such as trackand-field events, bicycle racing, and swimming. In each case, explain how the power is determined. Make a table to compare the various power outputs.

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