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Chapter 8: Problem 66

A jet aircraft is traveling at \(223 \mathrm{~m} / \mathrm{s}\) in horizontal flight. The engine takes in air at a rate of \(80.0 \mathrm{~kg} / \mathrm{s}\) and burns fuel at a rate of \(3.00 \mathrm{~kg} / \mathrm{s}\). The exhaust gases are ejected at \(600 . \mathrm{m} / \mathrm{s}\) relative to the speed of the aircraft. Find the thrust of the jet engine.

Short Answer

Expert verified
Answer: The thrust of the jet engine is 50,369 N.

Step by step solution

01

Identify the mass flow rates

We are given the mass flow rates for the intake air \(m_{air}\) and the burning fuel \(m_{fuel}\): \(m_{air} = 80.0\,\mathrm{kg/s}\) \(m_{fuel} = 3.00\,\mathrm{kg/s}\) Now, we determine the mass flow rate of the exhaust gases, \(m_{exhaust}\), which is just the sum of the intake air and burning fuel mass flow rates: \(m_{exhaust} = m_{air} + m_{fuel}\)
02

Calculate the mass flow rate of the exhaust gases

Using the given mass flow rates, we calculate the mass flow rate of the exhaust gases: \(m_{exhaust} = 80.0\,\mathrm{kg/s} + 3.00\,\mathrm{kg/s} = 83.0\,\mathrm{kg/s}\)
03

Determine the relative exhaust gas velocity

We are given the exhaust gas velocity relative to the aircraft's velocity \(u_{relative}\): \(u_{relative} = 600\,\mathrm{m/s}\) Keep in mind that the aircraft is traveling horizontally at a velocity \(u_{aircraft}\): \(u_{aircraft} = 223\,\mathrm{m/s}\) The actual exhaust gas velocity, \(u_{exhaust}\), will be: \(u_{exhaust} = u_{aircraft} + u_{relative}\)
04

Calculate the exhaust gas velocity

Using the given aircraft and relative exhaust gas velocities, we calculate the actual exhaust gas velocity: \(u_{exhaust} = 223\,\mathrm{m/s} + 600\,\mathrm{m/s} = 823\,\mathrm{m/s}\)
05

Determine the thrust of the jet engine

The thrust of the jet engine, \(T\), is determined by applying the conservation of linear momentum principle to the engine system: \(T = m_{exhaust}\times u_{exhaust} - m_{air}\times u_{aircraft}\) Plugging in the values, we get: \(T = (83.0\,\mathrm{kg/s})(823\,\mathrm{m/s}) - (80.0\,\mathrm{kg/s})(223\,\mathrm{m/s})\)
06

Calculate and report the thrust

Performing the calculations, we find the thrust of the jet engine: \(T = 68209\,\mathrm{N} - 17840\,\mathrm{N} = 50369\,\mathrm{N}\) Thus, the thrust of the jet engine is \(50369\,\mathrm{N}\).

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