Two identical balls, A and B, are projected with the same initial...

For ball A, the angle of projection with the horizontal is $\theta_A = 30^\circ$.

Speed of ball A at the highest point is its horizontal component of velocity:
$v_A = u \cos 30^\circ = u \left(\dfrac{\sqrt{3}}{2}\right)$

For ball B, the angle of projection with the vertical is $30^\circ$, so its angle with the horizontal is $\theta_B = 90^\circ - 30^\circ = 60^\circ$.

Speed of ball B at the highest point is:
$v_B = u \cos 60^\circ = u \left(\dfrac{1}{2}\right)$

The ratio of their speeds at the highest points is:
$\dfrac{v_A}{v_B} = \dfrac{u \left(\dfrac{\sqrt{3}}{2}\right)}{u \left(\dfrac{1}{2}\right)} = \dfrac{\sqrt{3}}{1}$

Therefore, the ratio is $\sqrt{3} : 1$.

Answer: $\sqrt{3} : 1$