From the video description

The F-35C took its first ride down a ship-borne catapult today as part of its ongoing inaugural sea trials aboard the USS Nimitz.

PACIFIC OCEAN (Nov. 4, 2014) F-35C Lightning II carrier variant Joint Strike Fighters launch from the aircraft carrier USS Nimitz (CVN 68) off the coast of San Diego. The launches and recoveries are part of initial at-sea Developmental Testing I (DT-I) for the F-35C, which commenced Nov. 3 and is expected to last two weeks. • The F-35C is the carrier variant (CV) of the F-35 Lightning II Joint Strike Fighter designed for the U.S. Navy as a first-day-of-war, survivable strike fighter complement to the F/A-18E/F Super Hornet. (U.S. Navy video/Released)

The second video is the arrested landing from Nov 3.