“In just 15 months, he built a life with us that was rich in friendship and shared adventure,” Dr. Reif said. “And he touched people across our community with his deep kindness and openhearted willingness to help, his humor and enthusiasm, his playful charm.”

The memorial’s speakers, who included Mr. Biden and Senator Warren, struggled to comprehend the circumstances around Officer Collier’s death. He is thought to have been sitting in his cruiser when the Tsarnaev brothers, Tamerlan and Dzhokhar, approached his car from behind and shot him, the authorities believe.

“It is so difficult to understand why such a senseless, brutal act was perpetrated on such a gentle, caring young man,” said the M.I.T. police chief, John DiFava.

During an emotional address, Mr. Biden drew on his own experience of losing a child, and reflected broadly — and colorfully — on what had motivated the bombers to commit their terrorism.