Defying Gravity! - 4-sided

This is an extension of this problem.

Now, imagine four surfaces of equal length making a square in a controlled environment, with same parameters (the gravity can be switched to be perpendicular to any of the surfaces). Now, say a ball is dropped from the midpoint of one of the sides of the square, falling in direction perpendicular to the adjacent side of the square. After a certain time period \(t_1\), the direction of the acceleration due to gravity is switched in a cyclic manner, to the next adjacent surface. Then, after time \(t_2\), the direction is switched in the same fashion. Then after every time period \(T\), the direction of gravity is changed. The periodic changes occur in such a way that the ball visits the midpoints of all four surfaces. If the surfaces are \(1 km.\) long, and the acceleration due to gravity is \(10 ms^{-2}\), find \(\lfloor t_2-t_1\rfloor \).

Also, derive the expression for \(T\), in terms of \(k\) and \(g\), which are length of surface and acceleration due to gravity respectively.

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