Airplane Takeoff Suppose that the distance an aircraft...

Airplane Takeoff Suppose that the distance an aircraft travels along a runway before takeoff is given by $D=(10 / 9) t^{2},$ where $D$ is measured in meters from the starting point and $t$ is measured in seconds from the time the brakes are released. If the aircraft will become airborne when its speed reaches 200 km%h, how long will it take to become airborne, and what distance will it have traveled by that time?

Key Concepts

Uniformly Accelerated Motion

This concept refers to motion in which an object experiences constant acceleration. When displacement is expressed as a quadratic function of time—for example, a function proportional to t²—it indicates that the object's acceleration does not change over time. Uniformly accelerated motion is fundamental in kinematics, as it allows us to predict velocity and position at any given time using simple mathematical relationships.

Differentiation in Kinematics

Differentiation is used to find the instantaneous velocity of an object by taking the derivative of its displacement function with respect to time. In problems involving motion, this technique is essential for transitioning from a position-time relationship to a velocity-time relationship, thereby enabling the determination of when a specified speed is achieved.

Unit Conversion

Unit conversion is critical in physics to ensure consistency when working with different measurement systems. In kinematics, it is common to convert between units such as kilometers per hour and meters per second. Correctly converting these units is vital to accurately apply equations and compare quantities across calculations.

Algebraic Problem Solving

This involves setting up equations based on physical relationships and then manipulating them to solve for unknown quantities. In the context of motion, algebraic techniques are used to isolate variables—such as time or distance—once the relevant relationships among acceleration, velocity, and displacement have been established. This skill is central to solving many problems in analytical mechanics.

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