A ball with a speed of 9 m s$$^{-1}$$ collides with another identical ball at rest. After the collision, the direction of each ball makes an angle of 30° with the original direction. If the ratio of the velocities of the balls after the collision is $$x : y$$, then what is the value of $$x$$?

We have a ball moving with speed 9 m/s colliding with an identical ball at rest. After collision, each ball makes an angle of 30° with the original direction.

Let the velocities after collision be $$v_1$$ and $$v_2$$. We apply conservation of momentum along the original direction (x-axis) and perpendicular to it (y-axis).

Along the x-axis: $$m \times 9 = m v_1 \cos 30° + m v_2 \cos 30°$$.

This simplifies to $$9 = (v_1 + v_2)\cos 30°$$.

Along the y-axis: $$0 = m v_1 \sin 30° - m v_2 \sin 30°$$.

This gives us $$v_1 \sin 30° = v_2 \sin 30°$$, so $$v_1 = v_2$$.

Since the two balls have equal velocities after collision, the ratio $$v_1 : v_2 = 1 : 1$$.

Hence, the value of $$x = 1$$.

So, the answer is $$1$$.