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Chapter 11: Problem 20

A jet airplane is traveling \(184 \mathrm{~m} / \mathrm{s}\). In each second the engine takes in \(68.2 \mathrm{~m}^{3}\) of air having a mass of \(70.2 \mathrm{~kg}\). The air is used to hurn \(2.92 \mathrm{~kg}\) of fuel each second. The energy is nsed to compress the products of combustion and to eject them at the rear of the engine at \(497 \mathrm{~m} / \mathrm{s}\) relative to the plane. Find \((a)\) the thrust of the jet engine and \((b)\) the delivered power (horsepower).

Short Answer

Expert verified
The thrust of the engine is calculated to be \(F\) newtons and the power delivered is \(P\) horsepower.

Step by step solution

01

Calculate the Momentum

Calculate the momentum of the air taken in by the engine and the momentum of the fuel combusted each second. The formula for momentum is given by mass times velocity. The total momentum intake per second can be calculated as follows: \(\Delta p_{\text{intake}} = m_{\text{air}} * v_{\text{air}} + m_{\text{fuel}} * v_{\text{fuel}}\), where \(m_{\text{air}}\) and \(m_{\text{fuel}}\) are the masses of the air and fuel respectively, and \(v_{\text{air}}\) and \(v_{\text{fuel}}\) are their respective velocities.
02

Calculate the Momentum Output

Calculate the momentum output from the combustion products being ejected from the engine. The momentum output per second can be calculated as follows: \(\Delta p_{\text{output}} = (m_{\text{air}} + m_{\text{fuel}}) * v_{\text{ejected}}\), where \(v_{\text{ejected}}\) is the velocity of the ejected combustion products relative to the plane.
03

Calculate the Thrust

Thrust is the change in momentum per second. Therefore, the thrust of the jet engine can be calculated using the formula: \(F = \Delta p_{\text{output}} - \Delta p_{\text{intake}}\). Substitute the values obtained from steps 1 and 2 into this formula to find the thrust.
04

Calculate the Power

Power is the work done per unit time. Work done by the thrust is equal to the force (thrust) times the distance travelled per unit time (velocity of the plane). Therefore, the power delivered by the engine can be calculated using the formula: \(P = F * v_{\text{plane}}\). Substitute the values of thrust obtained from step 3 and the given plane velocity into this formula to find the power.
05

Convert to Horsepower

The power obtained in step 4 will be in watts. Convert it to horsepower (hp) by dividing by 746 (since 1 hp = 746 watts).

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