Defying Gravity! - 3-sided

This is an extension on this problem

Consider, now, three surfaces of equal length forming an equilateral triangle in a controlled environment. Just like the previous problem , the gravity of the system can be altered to be perpendicular to any one of the surfaces. Now, consider a ball being dropped from the midpoint of one of the sides, falling perpendicular to one of the sides. Now, after a certain time $t_1$ , the direction of the acceleration due to gravity is switched such it is perpendicular to the third surface of the triangular set up. After another time period, $t_2$ , the gravity is switched again in a similar fashion, then after every time period $T$ , the gravity is switched in this manner. The switching of the gravity causes the ball to touch the midpoints of the three surfaces. If the length of one of the surfaces is $100 m.$ , and the acceleration due to gravity is $10 ms^{-2}$ , find $t_1$ , to three decimal places.

Also, derive formulas for $t_2, T$ , in terms of length of surface $k$ , and acceleration due to gravity $g$ .

Need a bigger challenge? Try this harder version of the problem.