Water leaks out of a small hole in the side of a bucket. The hole is a distance \(d\) below the surface of the water, and the diameter of the hole is much smaller than the diameter of the bucket. Example 18. 9 derived an expression for the speed \(v_{2}\) at which water exits the hole by assuming that \(v_{1} \approx 0\) for the speed at which the water surface moves downward. Derive an expression for \(v_{2}\) when \(v_{1}\) cannot be ignored.

Data from Example 18. 9

Water leaks out of a small hole in the side of a bucket. The hole is a distance \(d\) below the surface of the water, and the cross section of the hole is much smaller than the diameter of the bucket. At what speed does the water emerge from the hole?