Why was the bay of pigs an embarasment to the United States?

The main appeal of the Bay of Pigs as an invasion point is that it is remote. It is, other than that, overly shallow. The USN destroyers which were the main U.S. support of the invasion could not get in close to bring their guns into action or to rescue the invasion force if it was discovered. It was discovered almost immediately and was swarmed by Cuban volunteers and Army. The US supplied no air cover and the supporting supply ships were sunk by the small Cuban Air Force.

Bad planning, limited resources and an alert and resourceful enemy doomed the invasion. There was supposed to a general uprising in response to the invasion that never developed. The event solidified Castro's control on the country. It also led to the Cuban missile crisis. Internationally and domestically it was a black eye for Kennedy.

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the constitution of the united states gave congress the absolute authority over immigration laws