Memory Alpha
Memory Alpha

Captain Janeway and Seven of Nine play Velocity

Velocity was a one-on-one competitive sport that was a cross between handball and a phaser range and held on a holodeck. Each player must avoid being hit by a disk by shooting it and sending it back toward their opponent. It was fast-paced game of wit and physical endurance, with both players in constant motion, ducking and spinning.

The disk used in the sport was holographically created, and was described as an aerodynamic, metallic object, eight inches across, glowing different colors when hit by each competitor's hand phaser – usually from red to blue, corresponding with the competitors' assigned colors. The game was played in rounds, and was concluded when a player achieves a "full impact" of the disk with his/her opponent's body.

Ensign Harry Kim was the captain of his Velocity team at Starfleet Academy. (VOY: "The Disease")

Lieutenant Reginald Barclay may have been unaware of this, since a holographic simulation of Kim that Barclay created wanted Barclay to teach him to play Velocity. (VOY: "Pathfinder")

Captain Janeway and Seven of Nine played a ten round game of Velocity in late-2374. Janeway won, six to four. Seven was annoyed, as she believed that her superior visual acuity and stamina should have easily allowed her to best Janeway, but Janeway described the game as being not just a test of stamina, but rather a game of wits. After their encounter with Arturis, Janeway and Seven played at least two more games. (VOY: "Hope and Fear")

When Janeway secluded herself from the crew of the USS Voyager while crossing the Void in 2375, Chakotay asked her to join him for a few rounds of Velocity on the holodeck. (VOY: "Night")

Beckett Mariner and Carol Freeman played a game of Velocity when the USS Cerritos had some downtime on its way to the Kontaran system in 2381. Brad Boimler attempted to join them, but hastily retreated after he realized they were talking about personal issues. (LD: "wej Duj")

It is unknown what setting a phaser is at when playing Velocity, but it is expected for the safety of the players that each phaser would be set to the lowest possible setting, or may possibly be a holographic simulation of a phaser, as phaser fire is known to disrupt holomatrices.
The outfits that Janeway and Seven of Nine wear while playing Velocity in "Hope and Fear" were evidently meant to match the Velocity disc's alternating colors of red and blue.
The Star Trek: Starfleet Academy novel The Assassination Game features a scene where James T. Kirk plays the game with Lartal, a Varkolak, suggesting that in the alternate reality, velocity existed as early as 2255.