Question

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

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

Step-by-step solution

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.

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 \( m \).

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 \( m \).

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.

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 \( t = 2.0 \text{ s} \).

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 \( 3.8 \times 10^7 \text{ watts} \).

Key concepts

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.