Question

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

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

Step-by-step solution

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.

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: \[ m = \frac{2W}{v^2} = \frac{2 \times 7.6 \times 10^7}{72^2} = 2939.5 \text{ kg} \]

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: \[ P = \frac{7.6 \times 10^7}{2.0} = 3.8 \times 10^7 \text{ W} \]

Key concepts

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.