Knockback is a measure of how far an attack sends its target. For example, Bowser's forward smash is an attack of high knockback; it launches opponents very far, so far that at high damages it's more powerful than a Home-Run Bat's smash attack. Pikachu's neutral attack, on the other hand, has very low knockback - it hardly sends opponents anywhere, even at ludicrous damages.
The knockback of most attacks increases as the target's damage increases. Knockback is also affected by the target's weight, gravity, and a few other conditions (such as type effectiveness). The damage dealt by an attack is a significant factor in how much knockback it deals, but it is not the primary factor - this is the reason stale moves have less knockback than fresh ones, and why many chargable special moves deal less knockback when fully charged (the knockback values of the fully-charged move are altered to compensate for the higher damage, whereas the same is often not true when not fully charged).
Each hitbox of a move has two knockback values: a base knockback and a knockback scaling (also known as knockback growth). Base knockback is the minimum amount of knockback the attack can deliver (in normal circumstances), and knockback scaling is a factor that controls how much the knockback increases as damage increases. Moves with high base knockback deal high knockback under any circumstances, such as the swing of a Home-Run Bat. They additionally tend to be more effective at KOing when less knockback is needed to KO (such as against lighter characters, when near the edge of the stage, or when an opponent is handicapped to sustain more knockback). On the other hand, moves with high knockback scaling take less damage to reach KO potential, such as Luigi's forward smash. They additionally tend to be more effective at KOing when the opponent requires more knockback to be KO'd (such as against heavier characters, when far away from a stage's blast line, or when an opponent is handicapped to sustain less knockback). Advanced techniques to extend survival such as DI and momentum cancelling are also less effective the stronger a move's knockback scaling is.
In Melee, the highest knockback delivered and received by each character is given in match results, though the number lacks meaning. In Brawl, the velocity applied (in units per 1000 frames) is provided instead of the knockback value (though it's simply knockback divided by 0.03). The unit is given as "mph"; while this would presumably mean "miles per hour", this is obviously not the case, as a 1,000 mph hit barely sends characters anywhere. The following table gives an idea of various knockback strengths:
Brawl uses a second formula to alter knockback based on the target's gravity:
The result of this formula is added to the vertical component of the knockback. As a result, characters with higher gravity take more knockback then those with lower gravity, depending on how vertical the angle is.
If the move has a weight-based knockback value set (so the move deals set knockback), then d is set to that value, and p is always 10.
Other physics of knockback
Set knockback, sometimes referred to fixed knockback, is a property of some attacks where the knockback dealt by the attack does not depend on the opponent's percentage or how much damage the attack does. This results in it dealing the same amount of knockback regardless of stale-move negation.
List of set knockback moves
In Melee and Brawl there are attacks that deal zero knockback. These attacks normally deal damage to the foe but do not cause flinching, meaning that the character being hit can still perform any actions while they are being attacked. Some of these attacks can be used for racking up damage quickly, as they can hit foes repeatedly without causing flinching or sending them away. These moves can also be used to "steal" KOs from opponents in free for all battles, as while they do not cause knockback, they will take ownership of the KO on an already fatally struck opponent. Fox's Blaster is an infamous example of an attack with no knockback. Due to the fact that it does not deal knockback, it will not interrupt an opponent's moves or force them out of a punishable state such as being asleep or being stuck in the ending lag of a missed Rest. Fox's Blaster in Brawl can also turn idle foes around it they are shot repeatedly.
In Melee, phantom hits damage foes, but without any knockback. It is possible for any character to receive no knockback from attacks by certain factors, such as by being giant and metal simultaneously on a low damage ratio battle, or during the actions of certain attacks in Brawl, such as Charizard during the first frames of Fly. This is known as launch resistance. Additionally, if Master Hand or Crazy Hand is used on a battle by glitches or hacks in Melee, when he grabs a foe and then crushes it, other foes near him will still take damage, but with no knockback.
List of moves with no knockback
Knockback stacking is the means by which the game applies knockback to a character who has already been sent flying. The new knockback will sometimes replace the old knockback entirely, sometimes have no effect at all, and sometimes a merge will be performed between the two velocities. This has a large effect on combos, as the character being comboed will often have already been struck when another hit lands.
In Melee, the method of knockback stacking used varies based on how many frames separate the first and second hit in addition to the relative angle between the two knockback directions. When the two hits are separated by fewer than 10 frames, the second hit's knockback always replaces the first's. This prevents attacks that hit repeatedly from accelerating the character that they hit with each strike, which would be the case were they added. This can also lead to some interesting scenarios that allow strong hits to be canceled into weaker ones, although this is mostly prevalent in team battles and with certain items. If the two hits are separated by 10 or more frames, the knockback is either set to the stronger value or the two are merged. The relative angle between the two hits' launch angles comes into play here: if the angle is large enough (meaning the hits are in opposite directions), then the two hits are merged via vector addition. However, if the relative angle is small (meaning the hits are in the same direction) the stronger hit is used and entirely replaces the weaker one. Since merging occurs when the hits are in opposite directions, merged knockbacks will often be in a direction completely unrelated to either of the previous two. For example, if a character is struck up and right by a strong hit and then up and left by an equally strong hit more than ten frames later, the character will be sent straight up (as the horizontal values cancel out and the vertical ones stack). This often leads to there being residual knockback from a previous hit in a combo which must be taken into account when trying to follow up.
In addition, hitstun duration is always refreshed by the second hit regardless of the time and angle between them, and the amount of hitstun applied is the amount that would be applied if the opponent had not already been hit. (I.e., the additional knockback put on the opponent from any merging does not factor in to the equation used to calculate the amount of hitstun that should be put on the opponent.)
In Brawl, the second hit always completely replaces the first hit's direction unless the first hit was much stronger, in which case the direction does not change. No merging is ever performed.
The following is a chart demonstrating how much knockback each character sustains in Brawl. While weight is the primary attribute for determining how much knockback a character sustains, there are other attributes that factor in, such as a character's gravity. This results in some characters sustaining more knockback than lighter characters.
The knockback value shown is how much knockback a character sustains when hit with Marth's Critical Hit at 0%, arranged from least knockback sustained to most knockback sustained.