A Flight Instructor in a DA40 reported the door came open during takeoff and disconnected from the aircraft during climb. They returned to the departure airport.

Narrative

We were going to do a normal flight to practice some short field landings at ZZZ1 departing from ZZZ. Pre-flight and everything was normal. We were slightly rushing to complete the flight as we didn't want to get home past XA:30. Got on the plane and started it up everything was normal. I handled talking to ATC while the student completed the Engine run-up. No annunciator lights were noticed by the student or me during startup; taxi; or takeoff roll. As we were well into our takeoff roll; I noticed a sound coming the back of the cockpit and quickly turned around to try and grab the door before it flew open. I was not strong enough to hold the door down and it eventually flew open disconnecting from the fuselage shortly after rotation. After realizing the door was gone; I quickly took control of the aircraft and proceeded to land the plane on [Runway] XXL exiting the runway at [Taxiway] 1. After clearing the runway; we returned to the hangar and had the door delivered to us. Upon reflecting on the incident; I believe I made all the right decisions to maintain the safe operation of the Aircraft without taking on unnecessary risks. The only injury was a slight scratch to my thumb. Given the events happened so quickly it was hard to gather enough thoughts to make a well-informed decision but returning to the runway was the best course of action at the time.