Technically, a "free kick" is any kick that is not a scrimmage kick. In other words, any time the ball is put into play by kicking, and the ball is not snapped on a normal scrimmage down, it is a free kick. This includes kickoffs, as well as kicks following a safety or fair-catch.
The kick following a safety is unique because it is the only type of free kick where the kicker has the choice of punting the ball or kicking it off a tee. (In fact, they could dropkick it as well.) Because of this, many people mistakenly think that "free kick" means the kicker can choose how to kick the ball, when in fact that is only true of the free kick after a safety.
The normal rules for kickoffs apply to all free kicks: The ball must travel 10 yards before the kicking team can recover it (unless first touched by a receiver), and it must not go out of bounds.
In the NFL, the team receiving a punt has the opportunity to kick a field goal on the play immediately after a fair catch, even if there is no time remaining on the clock. The team attempting the free kick must either drop kick the ball or have a player holding the ball like on a 'regular' field goal attempt. A kicking tee is not allowed. The defensive team must be positioned a minimum of 10 yards from the line of scrimmage at the time of the kick. If the kicked ball goes through the uprights, it is considered a field goal and three points are awarded to the kicking team. If the attempt is no good, the defensive team is awarded the ball at the spot of the kick unless the defensive team attempts to return the kick. The free kick after fair catch play is rarely seen and is always done at the end of a half. There is no free kick after fair catch rule in college ball.
Following possession of a kickoffWhen a team has forced the other team to punt away, and calls a fair catch, they are entitled to take a free kick (fair catch kick in the NFL). This is basically a kick off the ground from where you caught the fair catch. This is different from any other kick because there is no snap and the defense does not rush the kicker.This is only of use if there is little or no time remaining and the ball is caught within field goal distance, which would be the intent of the kick. The kick is taken from the yard line of the catch, not from behind a line of scrimmage.Following a safetyA team that gives up a safety delivers a free kick from its own 20-yard line, either from a punt, a placekick without a tee, or a dropkick. A safety scores 2 points and possession from the free kick.
Following a safety, the ball can be put in play either by placekick, punt or dropkick. The kicking team may choose which method to use. In college and high school, the kicker may use a tee if he opts to placekick the ball. However, in the NFL a tee is not allowed for a free kick following a safety, so the team would have to employ a holder. Note that these rules only apply to a free kick following a safety. Contrary to popular belief, this is NOT the definition of a free kick in general. Any kick which is not a scrimmage kick is a free kick, including kickoffs and fair-catch kicks.
The team who the safety was against has to kick off to the team that scored the safety (free kick) from the 20 yard line. A free kick means that they can kick off using a tee, or they may punt the ball.
By rule, if you kick the ball during a scrimmage down, it's called a "scrimmage kick." If a scrimmage kick off the ground goes through the uprights, it's a field goal. Technically, there is no rule that defines a field goal "attempt." You don't necessarily have to "attempt" a field goal when placekicking on a scrimmage down, but if your kick doesn't go through the uprights, the same rules apply as would for a missed field goal.
See Web Links for NFL Rule book
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they kick the football in the case of a kickoff, punt, turnover or field goal kick.