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.
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.