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 49,700 N.

Step-by-step solution

Determine the mass flow rate through the engine

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 total mass flow rate through the engine is the sum of these two quantities: \(mass\_flow\_rate = air\_mass\_flow\_rate + fuel\_mass\_flow\_rate = 80.0 + 3.00 = 83.0 \mathrm{~kg} / \mathrm{s}\).

Calculate the initial momentum per unit time

The initial momentum per unit time is the momentum of the incoming air and fuel mixture. To calculate it, multiply the mass flow rate by the speed of the aircraft: \(initial\_momentum\_per\_unit\_time = mass\_flow\_rate \times speed = 83.0 \times 223 = 18509 \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s}^2\).

Calculate the momentum change per unit time

The exhaust gases are ejected at the speed of \(600 + 223 = 823 \mathrm{~m} / \mathrm{s}\) relative to the ground (since they are given relative to the aircraft speed). The change in momentum per unit time can be calculated by multiplying the mass flow rate by this speed: \(momentum\_change\_per\_unit\_time = mass\_flow\_rate \times exhaust\_speed = 83.0 \times 823 = 68209 \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s}^2\).

Determine the thrust

Now we can use the conservation of momentum principle to determine the thrust. The thrust is the change in momentum per unit time, so: \(thrust = momentum\_change\_per\_unit\_time - initial\_momentum\_per\_unit\_time = 68209 - 18509 = 49700 \mathrm{~N}\). The thrust of the jet engine is 49,700 N.

Key concepts

Conservation of Momentum

Understanding the conservation of momentum is vital when calculating quantities such as the thrust of a jet engine. Simply put, momentum, a quantity of motion, is conserved in a closed system where no external forces are applied. That is, the total momentum within the system before any event must equal the total momentum after the event.

For a jet engine propelling an aircraft, the system consists of the aircraft and the gases that pass through the engine. Air and fuel enter the engine, get compressed and combusted, and then expelled at high speed. Throughout this process, although the form of the substances changes, the momentum is transferred from the engine to the exhaust gases, propelling the aircraft forward. The conservation of momentum principle allows us to equate the momentum inflow to the momentum outflow plus the net external force, which in this case, is the thrust generated by the engine.

Momentum Change per Unit Time

Momentum change per unit time, also referred to as the rate of change of momentum, is essentially another expression for 'force'. This concept is crucial for understanding how engines generate force to produce motion.

When a jet engine expels exhaust gases at a greater velocity than the intake air, it enhances the aircraft's momentum over time; thus, there's a momentum change per unit time. Mathematically, we express this as the product of the mass flow rate and the difference in velocities of the exhaust and the intake air. By calculating the momentum change per unit time, one can directly find the thrust, as force equals the rate of change of momentum according to Newton's second law of motion. In the case of our jet engine, a higher exhaust velocity results in a greater momentum change per unit time, yielding more thrust.

Mass Flow Rate

The mass flow rate is a measurement of the amount of mass passing through a given surface per unit time. It's a key concept in fluid dynamics and plays a pivotal role in calculating jet engine thrust. For engines, the mass flow rate is the mass of air and fuel that passes into and out of the engine per second.

Knowing the mass flow rate allows us to determine how much momentum is being carried by the air and fuel mixture into the engine and how much is being expelled as exhaust gases. These two rates help us to calculate the change in momentum which, when multiplied by the relative exhaust speed, results in the thrust. For example, an engine with a high mass flow rate can handle more air and fuel, typically generating greater thrust, assuming efficiency and other factors remain consistent.