A ball of mass mm moving with velocity v_i strikes a vertical wall as shown in the figure. The angle between the ball's initial velocity vector and the wall is ?_i as shown on the diagram, which depicts the situation as seen from above. The duration of the collision between the ball and the wall is ?t, and this collision is completely elastic. Friction is negligible, so the ball does not start spinning. In this idealized collision, the force exerted on the ball by the wall is parallel to the xaxis.

What is the final angle ?_f that the ball's velocity vector makes with the negative yaxis? Express your answer in terms of some or all of the variables m, v_i, ?_i, ?t.