https://en.wikipedia.org/wiki/US_Airways_Flight_1549
Once an engine breaks the question is whether or not it becomes unbalanced, that is one reason why they can become detached from the plane.
> In an effort to prevent similar accidents, officials captured and exterminated 1,235 Canada geese at 17 locations across New York City in mid-2009 and coated 1,739 goose eggs with oil to smother the developing goslings.[71] As of 2017, 70,000 birds had been intentionally killed in New York City through programs instituted after the ditching.[72]