A long-range jet liner flies (almost exactly) level at constant speed, subject to the forces of drag fD=(1/2)CDrhoSv^2, thrust Ft, and weight Mg. Given the plane's wing area, lift and drag coefficients, speed and takeoff mass, calculate the engine power required for level flight during the cruise phase of its long nonstop flight, as a function of distance traveled. Numerical evaluation of all coefficient in your result will be required.
To calculate the engine power required for level flight during the cruise phase, we need to consider the forces acting on the jet liner and the power equation.
The forces acting on the jet liner are:
1. Weight (Mg) - Due to the force of gravity acting on the mass of the aircraft.
2. Lift (L) - Generated by the wings to counteract the weight and keep the plane in the air.
3. Drag (fD) - The force opposing the motion of the aircraft caused by air resistance.
The power required for level flight can be calculated using the power equation:
Power = Force x Velocity
To determine the thrust required, we consider that in level flight, the total drag force must be equal to the total thrust force (Ft) provided by the engines:
fD = Ft
So, we can set the drag equation equal to the thrust:
(1/2)CDρSv^2 = Ft
Now, let's calculate each component:
1. Lift (L):
Lift can be calculated using the lift equation:
L = (1/2)CLρSv^2
Where:
- CL is the lift coefficient.
- ρ is the density of air.
- S is the wing area.
- v is the velocity of the aircraft.
2. Drag (fD):
Drag can be calculated using the drag equation:
fD = (1/2)CDρSv^2
Where:
- CD is the drag coefficient.
- ρ is the density of air.
- S is the wing area.
- v is the velocity of the aircraft.
Since fD = Ft, we can substitute fD in the power equation to calculate the engine power required for level flight:
Power = ft x v
Substituting ft with (1/2)CDρSv^2, we get:
Power = (1/2)CDρSv^3
Now, the engine power required for level flight during the cruise phase can be calculated as a function of the distance traveled by multiplying the power by the speed:
Power_required = Power x distance_traveled
Where:
- Power_required is the engine power required for level flight.
- Power is the calculated power based on the equations above.
- distance_traveled is the distance traveled during the cruise phase.
To evaluate the numerical coefficients, you'll need to input the specific values of the following parameters:
- CD (drag coefficient)
- ρ (density of air)
- S (wing area)
- v (velocity of the aircraft)
- distance_traveled
By substituting these values into the equations and evaluating the result, you can find the engine power required for level flight during the cruise phase as a function of the distance traveled.