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

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AI Thread Summary
The discussion centers on a pursuit problem involving two entities moving in a spiral towards the center of a square. The velocities of the entities are perpendicular and maintain a constant radial component directed towards the center, which remains unchanged as the square shrinks. The challenge lies in calculating the time it takes for them to meet, given that their velocities continuously change direction. The participants explore the concept of using average velocity to determine the time interval for a specific distance reduction. Overall, the problem emphasizes the geometric properties of the square and the dynamics of motion within it.
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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.
 
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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