Find the length of a side in a right triangle given one side and two angles.

1. Identify the problem type: This is a trigonometry problem involving a right triangle. We are given one side and two angles, and we need to find the length of another side.
2. Analyze the given information: We have a right triangle. One angle is $30^\circ$, another is $60^\circ$, and the third is $90^\circ$. The side opposite the $30^\circ$ angle has a length of $24$ mm. The side we need to find, labeled $u$, is opposite the $60^\circ$ angle.
3. Choose the appropriate trigonometric ratio: In a right triangle, the tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side. We can use the tangent of the $60^\circ$ angle to relate the side $u$ to the side with length $24$ mm. Alternatively, we can use the tangent of the $30^\circ$ angle.
Using the $60^\circ$ angle:
$$\tan(60^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{u}{24}$$
Using the $30^\circ$ angle:
$$\tan(30^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{24}{u}$$
4. Solve for $u$ using the $60^\circ$ angle: We know that $\tan(60^\circ) = \sqrt{3}$.
$$u = 24 \times \tan(60^\circ)$$
$$u = 24 \times \sqrt{3}$$
$$u = 24\sqrt{3}$$
5. Solve for $u$ using the $30^\circ$ angle: We know that $\tan(30^\circ) = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$.
$$u = \frac{24}{\tan(30^\circ)}$$
$$u = \frac{24}{\frac{1}{\sqrt{3}}}$$
$$u = 24 \times \sqrt{3}$$
$$u = 24\sqrt{3}$$
6. State the final answer: The length of side $u$ is $24\sqrt{3}$ millimeters. The answer is already in simplest radical form.