Minutes into a 30-minute doors-off sightseeing helicopter flight, passengers who had just been laughing and exchanging jokes struggled to free themselves from harnesses as the chopper sank in the East River, according to a newly released report.

“How do I cut this?” one of the five doomed passengers asks, 13 seconds before the aircraft goes under to the sounds of the rotors slicing the water, according to a transcript of video taken from a GoPro 5 camera mounted on the cabin ceiling.

The National Transportation Safety Board made the transcript public on Monday, but did not release the chilling video itself.

All five passengers drowned on March 11, 2018, when the Eurocopter AS350 flight booked by FlyNYON plunged into the East River and they became trapped in their safety harnesses.

The tragedy unfolded in mere seconds, according to the transcript, with one passenger reaching for his chest where a hook knife had been seen earlier, another pulling uselessly at the shoulder straps of his harness and a third heard panting.

Pilot Richard Vance — who had been wearing only a seatbelt — managed to bail out and was rescued from the chopper, which was owned and operated by Liberty Helicopters.

The agency also released a trove of other documents from its probe into the deadly crash and set a Dec. 10 hearing in Washington to determine a probable cause.

The chopper crashed after a tether meant to keep passengers safe got caught on a fuel shutoff switch, stopping the engine, Vance told investigators.

He said that once the engine stopped, he took action to glide the aircraft away from crowded areas — like tall buildings and Central Park — but hadn’t noticed the fuel switch was to blame until he looked down at it just before impact.

Asked by investigators if he considered going back to rescue his passengers, Vance said he wanted to, but thought that his clothing would weigh him down.

FlyNYON pointed to the shutoff as a “long standing design issue” with the floor-mounted fuel controls of the helicopter, as well as problems with its emergency flotation system, which failed to keep the aircraft from flipping over and sinking.

“These issues continue to pose a threat to passengers and crew on AS350B2 helicopters and any helicopter employing the same Dart emergency flotation system,” FlyNYON spokesman Jason Kaplan told The Post in a statement.

“We shared our concerns with the NTSB and urge them to alert the FAA at the highest levels to ensure they are aware of these very serious safety issues,” he added.

Liberty Helicopters did not immediately respond to a message seeking comment.

In addition to the GoPro camera, investigators collected 57 photographs and 17 videos from an iPhone X that one of the passengers in the back held during the flight, according to the NTSB.

Two of the four passengers sitting in the rear were heard joking about alcohol before takeoff.

“I was like do I pop into a liquor store and get something?” one was heard saying, according to the transcript. “Then I was like, ‘No,’ business meeting. Shouldn’t do. No shots before a business meeting.”

The pilot then said: “They tell us we’re not supposed to drink before we fly, too. But I mean, come on.” The transcript notes that he spoke “in a joking tone.”

Vance then told the passengers to leave their seatbelts on for the whole flight. “You’re tethered. You’re not going anywhere,” he said.

“Once I say you’re free to do your thing, go for it, you know you guys pay to dangle,” added Vance, who provided a safety briefing that included who was going to remove seatbelts, who would remain buckled and how to use seatbelt cutters, according to the report.

In the final moments of the crippled flight, Vance activated the chopper’s floats at an altitude of about 800 feet after trying unsuccessfully to restart the engine.

“At this point he was ‘committed to impact and reached down for the emergency fuel lever. He ‘could tell something was wrong because it was in the up position,'” according to a transcript of an interview with the pilot.

“By the time he unbuckled his seatbelt he was ‘fully under water’ and used two hands to grab the door frame and pull himself out,” it said. “The helicopter was ‘rolling on top’ of him but he was able to get to the surface. He surfaced about 4 feet away from the nose of the helicopter and crawled up onto the belly. He stood up and waved for help but could not see anything so he just waited.”

When asked by investigators about the emergency fuel shutoff lever, he said “it was in the up position” and he saw the front passenger’s “tether loop was underneath the handle.”

And when asked whether he had concerns about any of his flights, he answered that he was worried about “whether or not the tether could physically be cut by the knives carried by passengers.”

The matter had been discussed “verbally with people developing the SOP,” he said, referring to standard operating procedures. “When asked what the response was, he stated that they were ‘trying to get a better option.'”

Vance also was asked if he had ever been through a dunker training simulator. He said said he had not but had talked to other pilots who had undergone the training.

The Federal Aviation Administration grounded doors-off flights after the crash, but they’ve since resumed.

Sens. Charles Schumer (D-NY) and Bob Menendez (D-NJ) have called on the FAA to close a loophole that allows doors-off flights for aerial photography, saying it’s geared for professional and commercial purposes, not tourists.

“FlyNYON and their perilous doors-off flights remain an egregious danger in the sky to this day,” Schumer said in a statement.

The passengers who died were Dallas Fire Officer Brian McDaniel, 26; his friend Trevor Cadigan, 26, a journalist from Dallas who had recently moved to New York; Carla Vallejos Blanco, 29, a tourist from Corrientes, Argentina; Tristan Hill, 29, who for a time was a basketball operations assistant with the Westchester Knicks, a Development League affiliate of the New York Knicks; and Daniel Thompson, 34.

With Post wires