“Chasing Vertices: A Time-Bound Pursuit Problem in a Square”

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Homework Statement
Consider a square with a side length of 1 meter. At each vertex of the square, there is a point. Each point moves at a constant speed of 1 meter per second, and the direction of movement is always towards the next vertex. In other words, point 1 moves towards point 2, point 2 moves towards point 3, point 3 moves towards point 4, and point 4 moves towards point 1. How long will it take for all the points to meet at the center?
Relevant Equations
Whatever equations work. Was given to us while learning about vectors.
I know that the velocities are perpendicular to each other and that they are moving in a spiral and that they will meet in the centre of the square. From that, I know the displacement of the point but I do not know how to get the time, since the velocity is always changing direction. Could I somehow take the average velocity?
 
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The radial component (towards the center) of each velocity has the same magnitude during the spiraling towards the center.
 
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nasu said:
The radial component (towards the center) of each velocity has the same magnitude during the spiraling towrads the center.
The implication is that the square remains a square as it shrinks. (As was already clear).

The radial component is independent of scale. So it is constant over time.
 
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jansons said:
since the velocity is always changing direction
Two mice are one meter apart. By how much is that distance reduced in a time interval of ##dt## ? :wink:

[edit]
This one is known as the mice problem. See also Radiodrome

##\ ##
 
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FAQ: “Chasing Vertices: A Time-Bound Pursuit Problem in a Square”

What is the main objective of the "Chasing Vertices: A Time-Bound Pursuit Problem in a Square" problem?

The main objective of the "Chasing Vertices: A Time-Bound Pursuit Problem in a Square" is to analyze and solve the dynamics of a pursuer attempting to catch a target within a square boundary. The problem typically involves determining the optimal paths and strategies for both the pursuer and the target, taking into account constraints such as time and movement capabilities.

What are the key constraints and assumptions in this pursuit problem?

Key constraints and assumptions in the problem include the following: the pursuer and target are confined to the boundaries of a square, both entities can move at a maximum speed, the initial positions of the pursuer and target are known, and the goal is to either minimize the time it takes for the pursuer to catch the target or to maximize the evasion time for the target. Additionally, the movements are often assumed to be continuous and can be described using geometric or differential equations.

What mathematical tools are commonly used to analyze this problem?

Mathematical tools commonly used to analyze this problem include differential equations, game theory, optimization techniques, and computational simulations. Geometry and calculus are also frequently employed to model the movements and interactions between the pursuer and target. In some cases, numerical methods and algorithms are used to approximate solutions where analytical solutions are difficult to obtain.

How can the results of this problem be applied in real-world scenarios?

The results of this pursuit problem can be applied in various real-world scenarios such as robotics (e.g., autonomous drones or vehicles chasing targets), security (e.g., surveillance systems tracking intruders), and wildlife management (e.g., predators hunting prey). Understanding optimal pursuit and evasion strategies can enhance the efficiency and effectiveness of these systems.

What are some common variations of the pursuit problem in a square?

Common variations of the pursuit problem in a square include changing the shape of the boundary (e.g., a circle or polygon), introducing multiple pursuers or targets, adding obstacles within the boundary, varying the speed and movement capabilities of the pursuer and target, and incorporating probabilistic elements to account for uncertainty in movement. These variations can significantly alter the complexity and strategies involved in solving the problem.

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