Turn it back on, go to Settings/General/Battery. Tap the graph on top to get to the second page. If there's a sharp drop to 0 in that graph, the battery is shot. (See
Lithium Battery Dendrite links for details.)
Make sure it's not getting really hot somewhere (top back area, maybe) when it's shutting down.
Copy the video to a computer and play it - does it go crazy at the point that the phone has a problem with? Then it's the video.
(There are a lot of possibilities - it's not possible to narrow it down to one cause without having the phone in hand.)