3. In a right-angled triangle table with one acute angle of $30^{\circ}$ and a hypotenuse of $4 \mathrm{~m}$, there are pockets at each corner (the size of the pockets can be ignored). A ball is placed in front of the pocket at the $60^{\circ}$ angle, and it is hit towards a point on the opposite side (not a vertex). Assuming the force of the hit is sufficient, the ball, after multiple reflections, falls into the pocket at the $30^{\circ}$ angle. The distance the ball travels on the table is $\qquad$ m.
$3.4 \sqrt{13}$.
Construct rectangle $A B C D$ such that $A B=2 \sqrt{3} \text{ m}, B C=14 \text{ m}$, and divide rectangle $A B C D$ into several triangles congruent to the tabletop as shown in Figure 4.

Place the ball at point $B$, and strike it towards point $E$, which is point $D$. It is easy to see that the ball will eventually fall into the pocket at the $30^{\circ}$ angle represented by point $D$ after 8 collisions.
Thus, $B D=\sqrt{A B^{2}+B C^{2}}=4 \sqrt{13} \text{ m}$.