Thrust Required for Level, Unaccelerated Flight

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In level, unaccelerated flight, an aircraft maintains constant altitude and velocity. Under this condition, thrust is needed solely to overcome aerodynamic drag. The amount of thrust required depends on aircraft characteristics, atmospheric conditions, and flight speed.

1. Force Balance in Level Flight

In steady, level flight:

  • Lift equals Weight: L = W
  • Thrust equals Drag: T_R = D

So, the thrust required (T_R) is simply:

T_R = D = \frac{1}{2} \rho V^2 S C_D

Where:

  • \rho = Air density
  • V = True airspeed
  • S = Wing reference area
  • C_D = Drag coefficient

2. Drag Coefficient in Terms of Lift

Drag consists of:

  • Parasite drag (C_{D0}): Independent of lift.
  • Induced drag: Related to lift.

Using the drag polar:

C_D = C_{D0} + k C_L^2

Also, in level flight, lift equals weight:

C_L = \frac{2W}{\rho V^2 S}

Substituting C_L into the drag equation:

C_D = C_{D0} + k \left( \frac{2W}{\rho V^2 S} \right)^2

Now substitute C_D into the thrust required equation:

T_R = \frac{1}{2} \rho V^2 S \left[ C_{D0} + \frac{4kW^2}{\rho^2 V^4 S^2} \right]

3. Behavior of Thrust Required with Speed

The thrust required curve (as a function of speed) is U-shaped:

  • At low speeds, induced drag is high → more thrust required.
  • At high speeds, parasite drag is high → more thrust required.
  • There is a minimum thrust required at an intermediate speed.

4. Minimum Thrust Condition

To find the speed at which thrust required is minimum, differentiate T_R with respect to V and set the derivative to zero:

\frac{dT_R}{dV} = 0

This gives the condition for minimum drag speed (V_{min,T_R}), which also corresponds to maximum L/D ratio.

5. Thrust Required vs. Thrust Available

  • Thrust Required Curve: U-shaped, depends on drag.
  • Thrust Available: Depends on the engine type and decreases with altitude (for jet engines).

Intersection point → Speed where aircraft can fly level.

If thrust available > thrust required, aircraft can:

  • Climb (excess thrust)
  • Accelerate (excess power)

If thrust available < thrust required, level flight is not possible.

6. Summary

In level, unaccelerated flight, the thrust required equals the total drag. It varies with speed due to the opposing trends of parasite and induced drag. Understanding the thrust required curve is essential for analyzing performance, fuel efficiency, and determining safe and efficient cruising speeds.

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