Master and Commander.
The ship is rounding Cape Horn in a massive storm, and the captain orders the men to reef the sails on their jury-rigged mast so it won't be blown off. One of the crewmen gets stuck up there however, and the mast falls over into the sea, him with it. But the mast is still connected to the ship by the rigging, and is dragging against it, threatening to capsize it.
So the captain is faced with a choice: Risk the ship long enough for the crewman to swim to safety, or cut it away, sending him to his death and saving the ship? He elects to cut the rigging away, and the crewman drowns. Afterwards, there's an incredibly poignant scene where the crewman's friend is going through his things and breaks down crying.