Use integration to obtain an expression for the time interval needed for the water surface in the bucket in Problem 71 (or Example 18. 9) to drain down to the level of the hole. You can't ignore \(v_{1}\) anymore, but you can use either the expression for \(v_{2}\) obtained in Problem 71 or an approximate expression for \(v_{2}\) that is valid during most of the time interval in which the water is draining.

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?

Data from Problem 71

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.