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With regard to a knife wielding person and the time for the distance between the perp and the intended victim - the incident one year ago in the shopping mall in St. Cloud Minnesota is a good case to study.
Perhaps readers will recall the situation when the Muslim deranged man began stabbing people willy-nilly and running around yelling at them. A man in the mall who was an off duty police officer had his gun with him. He had a concealed carry permit. When the perp entered Macy’s department store he charged at the officer who warned him that he had a gun. The man continued unabated and was shot one, two, three and finally about five or six times before he stopped his attacking behavior and slumped to the floor and died. When questioned, the officer stated that he kept telling the perp to stop, but with no avail. An idiotic reported asked him why he had to shoot so many times. The officer stated that his purpose was to stop the attack and it was inevitable at that point that the man was shot in the head.
So, in answer to how and why an officer would finally shoot to kill a knife-wielding person, the most likely answer is - it was the only way to halt the attack.