Final answer:
The delivered power of a jet engine that takes in air at a rate of 112 kg/s and ejects exhaust gases at 450 m/s relative to the aircraft traveling at 217 m/s is approximately 5.66 MW.
Step-by-step explanation:
To find the delivered power of the jet engine, we use the principle of conservation of momentum and the definition of power. Power is the rate at which work is done, or energy is transferred. In the context of a jet engine, power can be calculated by considering the change in kinetic energy per unit time for the mass of air and fuel expelled from the engine.
First, we need to find the force exerted by the jet engine. This can be done using the formula: Force (F) = rate of change of momentum. Since the exhaust gases are being ejected at a certain rate and speed, we can write this as:
F = m' * (v_exhaust - v_aircraft)
where:
- m' is the mass flow rate of the exhaust (sum of air and fuel).
- vexhaust is the speed of the exhaust gases relative to the aircraft.
- vaircraft is the speed of the aircraft.
Next, we calculate the power delivered by the engine using the formula Power (P) = F * v_aircraft, where F is the force exerted by the engine and v_aircraft is the velocity of the aircraft.
Given:
- Mass flow rate of air: m' = 112 kg/s (since fuel's mass flow rate is much less compared, we can approximate the mass of the exhaust gas by the mass of the air alone)
- Velocity of aircraft: v_aircraft = 217 m/s
- Velocity of exhaust gases relative to aircraft: v_exhaust = 450 m/s
Now we compute:
F = 112 kg/s * (450 m/s - 217 m/s) = 112 kg/s * 233 m/s = 26104 N
Then:
Power (P) = 26104 N * 217 m/s = 5661568 W or approximately 5.66 MW
The delivered power of the engine is therefore approximately 5.66 megawatts (MW).