We all know the standard trolley problem by now. A runaway trolley with five people trapped on it is on a track that's going to send it off a cliff, killing everyone on board. There's a lever that can send it onto a side track, though, that will result in the trolley stopping safely, but there is a single person tied to the track who will be run over by the trolley and die.
Let's make it even more horrible. The person tied to the side track is the only one who can move the lever. If the person chooses to die to save the others, most moral systems would generally consider this a Good and Noble Thing. But sacrificing oneself is difficult; most people want very much not to die, and except in unusual circumstances it goes against biological survival instincts. If someone doesn't sacrifice themselves, is this a morally permissible act, given human frailty? We do demand that soldiers, even conscripted ones, be willing to perform acts of self-sacrificial bravery, and call those who refuse "cowards" and "deserters". Does the initial position of the lever matter? A person who does not throw a lever to divert the trolley onto himself might be a coward, but a person who throws a lever to divert the trolley away from himself and over a cliff might be a murderer.