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

[jansons]

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?

 

[nasu]

The radial component (towards the center) of each velocity has the same magnitude during the spiraling towards the center.

 

[jbriggs444]

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.

 

[BvU]

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 $dt$ ?

[edit]

Spoiler

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

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