Summary: Facing the problem importing M2TS files in Final Cut Pro, you can fix the issues via transcoding your M2TS video to FCP compatible format first.
I have M2TS files from Sony Handy Cam to MacBook Pro, when import the file to Final Cut Pro, just show "unkonw the file". In fact, Final Cut Pro 6/7/X have add the AVCHD .mts, .m2ts formats as the input video, but there are so many limits you are facing like, system requirements, video frame rate (as I know FCP still can’t handle 1080 60p footage well), whole AVCHD structure(FCP can't recognize single MTS/M2TS video). So if you want to edit your M2TS files with Final Cut Pro smoothly, you will need to transcode the .m2ts files to FCP supported format, Apple ProRes.
Now I share the guide with you how to convert M2TS footage to Final Cut Pro friendly format using Brorsoft MTS/M2TS Converter for Mac. With it, you can effortlessly encode/re-encode single M2TS footage shot by Canon, JVC, Sony, Panasonic etc. cameras for smoothly importing and editing in various non-linear editing systems (NLE), including Adobe Premiere Pro, iMovie, Avid Media Composer, Final Cut Express and more. Now, follow the instruction how to import M2TS files into Final Cut Pro.
1. Download, install and run the MTS to FCP Converter; click "Add Videos" icon to load your source .m2ts videos.
2. Click the format bar, and move mouse cursor to "Final Cut Pro > Apple ProRes 422 (*.mov)" as output format.
Some more helpful features of the app:
a. Settings- click to set video resolution(1920×1080/1440×1080/1280×720/720×480), bitrate(from 1Mbps to 20Mbps), frame rate (24p/30p)
b. Editor (next to "Add" icon)- click to set deinterlace, trim, crop, watermark, subtitles, etc.
3. Click "Convert" button to start M2TS to ProRes 422 MOV conversion on Mac OS X.
As soon as the encoding process finished, you can get the final output Apple ProRes files via clicking "Open" button effortlessly. The transcoded files now are ready for editing in Final Cut Pro (X).
Launch FCP, choose File > Import > Files… and choose the videos that you'd like to add for editing.
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Source:Converting M2TS to ProRes .mov for Final Cut Pro on Mac