# Problem 3

Contributed by Ivan Erofeev (The University of Edinburgh).

A small ball of mass $m$ is suspended on an in-extensible massless thread of length $l$. The thread is very thin, flexible, and has a breaking force limit of $T_b = 9mg$ , where $g$ denotes the free fall acceleration. In the initial moment the thread is horizontal as shown in the figure, and the ball velocity $v = v_0$ is directed downward. It is known that the thread breaks in one of the four positions shown in the figure with dashed lines, when the angle between the thread and the vertical is 60$^\circ$; friction is to be neglected. Find all the possible values for $v_0$ . (A solution is considered to be correct only if all the suitable values are algebraically expressed, and no extra/invalid values are included.)

Please submit the solution of this problem via e-mail to physcs.cup@gmail.com. First hints will appear here on 26th January 2020, but after the publication of the first hint, the base score is reduced to 0.9 pts. For full regulations, see the “Participate” tab.