Question

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

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

Step-by-step solution

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: \(m_{exhaust} = m_{air} + m_{fuel}\)

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: \(m_{exhaust} = 80.0\,\mathrm{kg/s} + 3.00\,\mathrm{kg/s} = 83.0\,\mathrm{kg/s}\)

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: \(u_{exhaust} = u_{aircraft} + u_{relative}\)

Calculate the exhaust gas velocity

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

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: \(T = (83.0\,\mathrm{kg/s})(823\,\mathrm{m/s}) - (80.0\,\mathrm{kg/s})(223\,\mathrm{m/s})\)

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 \(50369\,\mathrm{N}\).