The plane of mass 5 -Mg begins to take off from an airport...

The plane of mass 5 -Mg begins to take off from an airport when at a speed of $250 \mathrm{~km} / \mathrm{h}$. If it carries an additional load of mass $400 \mathrm{~kg}$, what must be its air speed before the plane begins to take off at the same angle of attack?

Key Concepts

Relationship Between Mass and Takeoff Speed

Since the lift required for takeoff must balance the weight of the aircraft and lift is proportional to v², any increase in mass—and thus weight—requires an increase in the takeoff speed. Specifically, if the mass increases, the new required speed is found by multiplying the original speed by the square root of the ratio of the new mass to the original mass, reflecting the square-root dependency inherent in the lift equation.

Weight and Its Impact on Performance

Weight, defined as the gravitational force on the mass of the aircraft, is a crucial parameter in determining takeoff performance. An increase in weight, such as by adding additional load, directly impacts the amount of lift that must be generated, and therefore, necessitates adjustments in the takeoff speed to maintain sufficient lift for a successful takeoff.

Lift Generation in Aerodynamics

Lift is the aerodynamic force that acts perpendicular to the relative airflow across the wing. It is generated through the interaction of the wing’s shape, the airspeed, and the air density, and is expressed by the equation L = 0.5 * ? * v² * S * C_L. The coefficient of lift (C_L) is primarily determined by the wing’s shape and the angle of attack, and when the angle is maintained constant during takeoff, C_L remains constant, making lift directly proportional to the square of the velocity.

Lifting Condition for Takeoff

For an aircraft to successfully take off, the lift force must equal the weight of the aircraft. This condition requires that the aircraft reaches a minimum speed, known as the takeoff speed, at which the aerodynamic lift overcomes the gravitational pull resulting from the aircraft's weight. Any increase in weight necessitates a re-calculation of this equilibrium to ensure safe takeoff conditions.

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