A plane from Denver, Colorado, (altitude 1650 meters) flies...

A plane from Denver, Colorado, (altitude 1650 meters) flies to Bismark, North Dakota (altitude 550 meters). It travels at $650 \mathrm{km} /$ hour at a constant height of 8000 meters above the line joining Denver and Bismark. Bismark is about $850 \mathrm{km}$ in the direction $60^{\circ}$ north of east from Denver. Find parametric equations describing the plane's motion. Assume the origin is at sea level beneath Denver, that the $x$ -axis points east and the $y$ -axis points north, and that the earth is flat. Measure distances in kilometers and time in hours.

Key Concepts

Parametric Equations

Parametric equations describe a path in space by expressing each coordinate as a function of a common parameter, often time. In the context of motion, they allow us to represent the position of an object continuously and conveniently, which is particularly useful when the object's path involves different components, such as horizontal and vertical displacements.

Coordinate Systems

Understanding the coordinate system is crucial when translating a real-world scenario into mathematical form. In this problem, a Cartesian coordinate system with defined east (x-axis) and north (y-axis) directions is used, and the origin is set at a specific point. This setup facilitates the decomposition of motion into its components along these axes.

Vector Decomposition

Vector decomposition involves breaking a vector into its components using trigonometry. When an object moves in a particular direction at a given angle, its displacement and velocity can be split into horizontal components using sine and cosine functions. This concept is important for analyzing motion in a plane when directions such as '60° north of east' are specified.

3D Motion

Three-dimensional motion requires accounting for changes in position along three axes, often labeled x, y, and z. The vertical (z) component is handled separately from the horizontal motion, and an altitude or height function is used to represent this component. Even when the z-coordinate follows a specific rule (like staying constant relative to some other reference), it is important to consider it in the overall representation.

Unit Conversion and Consistency

Converting units and ensuring consistency throughout the problem is essential, especially when different units are initially provided (for example, meters vs. kilometers). Here, altitudes provided in meters need to be converted to kilometers to match the horizontal distances and speeds, preventing any discrepancies or errors in the equations.

Kinematics and Constant Velocity

Kinematics encompasses the study of motion, including concepts of constant velocity where the speed and direction remain consistent over time. When the object moves with a constant speed in a given direction, its displacement can be obtained by multiplying the velocity by time. This direct proportionality simplifies the derivation of parametric equations for motion.

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