On seeing Patches begin to run toward her, Heather slows...

On seeing Patches begin to run toward her, Heather slows down, her speed being $3 x \mathrm{~km} / \mathrm{h}$, where $x$ is Patches' horizontal coordinate. Thus, when Patches is at $x=1$, Heather's speed is $3 \mathrm{~km} / \mathrm{h}$, but when Patches reaches her $(x=0)$ Heather is standing still. If Patches still runs at a constant speed of $8 \mathrm{~km} / \mathrm{h}$, analytically determine the equation $y(x)$ describing Patches' path and plot it. How far has Heather walked when Patches catches her? How far has Patches run? How long does it take Patches to reach Heather?

Key Concepts

Time Integration

Time integration involves calculating the time required for an event by integrating a differential relation between time and another variable, such as distance. In motion problems, it is used to determine the total time taken by an object to complete its trajectory by relating its speed and the incremental distances covered.

Arc Length Integration

Arc length integration is the process of determining the distance traveled along a curve by integrating the differential element of arc length. In the context of kinematics, it allows one to compute the actual path length traversed by an object from its differential change in position, given its trajectory.

Parametric Representation in Kinematics

Parametric representation is a technique where the coordinates of a moving object are expressed as functions of a parameter, often time. This approach is valuable in describing complex motions where the path, speed, and acceleration need to be analyzed simultaneously, allowing for separate yet interconnected expressions for each coordinate.

Pursuit Curve

A pursuit curve describes the path followed by a moving object when it is directed continuously towards another moving target. This concept arises in problems involving chases and is modeled by setting up differential equations that capture how the chaser adjusts its direction and speed in response to the target's position.

Separation of Variables

Separation of variables is a method used to solve differential equations by isolating the different variables on opposite sides of the equation. This method allows the integration of each side with respect to its own variable, leading to an expression that relates them. It is particularly useful when the rate of change of one variable is given explicitly as a function of another.

Differential Equations

Differential equations are used to model relationships involving rates of change. In problems where one variable depends on another—in this case, a speed depending on a position coordinate—setting up and solving a differential equation becomes the key technique to link the positions, velocities, and ultimately the trajectories of the moving objects.

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