The correct answer is option 2, John Kay.

Option 1: James Hargreaves is credited with inventing the spinning jenny, which was a machine for spinning cotton. This invention greatly increased the speed and efficiency of spinning, but it is not related to the flying shuttle.

Option 3: Richard Arkwright is known for inventing the water frame, a machine that could spin a strong, hard twist of cotton thread. Again, this invention is not related to the flying shuttle.

Option 4: Humphry Davy was a chemist and inventor, known for his discoveries in the field of electrochemistry. He did not invent the flying shuttle.

The flying shuttle was invented by John Kay, an English inventor, in 1733. The flying shuttle was a mechanical device that significantly increased the speed and efficiency of weaving. It allowed a weaver to throw the shuttle back and forth across the width of the loom by pulling a cord. This eliminated the need for a second person to manually pass the shuttle through the warp threads, allowing for faster weaving and larger pieces of fabric to be produced. The invention of the flying shuttle was a significant development in the textile industry and played a crucial role in the Industrial Revolution.