# If action equals reaction, how is it ever possible to win in martial arts?

In kick-boxing, when a fighter's leg hits an opponents leg, the outcome, based on Newton's 3rd law, should be the same for each fighter. It's not even important who kicked who, as in the moment of contact the attacker should feel more less the same as the defender.

Here is a catch: in most of situations different parts of the fighters' bodies collide: the attacker typically contacts the front of his leg with the defender's side. The front is harder. Is it hardness that makes the difference?

Some web pages inform me, that because of the 3rd law, a fighter should make powerful but very brief hits - retracting a kicking leg before it receives a reaction. But from what I know, if you don't feel a reaction, there was no action in the first place.

How is it ever possible to take advantage in martial arts hit and win?

• youtube.com/watch?v=397lM2aZZt4 Sep 29 '15 at 6:07
• If I slam my fist into your face, both my fist and your face will experience the same force from the impact. (An equal and opposite reaction.) Now what do you suppose will be hurt more by that impact? My fist? Or your face? ;-) Sep 29 '15 at 13:40

In kick-boxing, when a fighter's leg hits an opponents leg, the outcome, based on Newton's 3rd law, should be the same for each fighter. It's not even important who kicked who, as in the moment of contact the attacker should feel more less the same as the defender.

You are correct. As you noted, Newton's third law does indeed say that the force on each fighter's body is the same (but in opposite direction) at each instant in time. This guarantees not only that the forces are equal, but also the impulse delivered to each of the colliding objects. Denoting the impulse imparted to objects 1 and 2 as $J_1$ and $J_2$ we therefore have

\begin{align} F_1(t) &= -F_2(t) \\ \text{and} \qquad J_1 \equiv \int dt \, F_1(t) &= -\int dt \, F_2(t) \equiv -J_2 \end{align}

Impulse has the same dimensions as momentum, so really what we're saying is that in a collision both objects experience the same change in momentum over the same period of time.

Some web pages inform me, that because of the 3rd law, a fighter should make powerful but very brief hits - retracting a kicking leg before it receives a reaction. But from what I know, if you don't feel a reaction, there was no action in the first place.

Again, you are absolutely right. Retracting the arm or leg does not reduce the force or impulse delivered to that arm or leg during the blow.

How is it ever possible to take advantage in martial arts hit and win?

You actually already partially got it when you mentioned hardness of the objects involved in the collision.

The technical words for describing this are stress and strain. Stress is essentially the inter-molecular or inter-atomic forces within a solid. Strain is the deformation of the solid from its usual shape.

When an arm or leg hits a nose, both experience the same impulse, but because the nose is softer, it deforms more. Once the nose tissues move too much relative to one another the the nose breaks. The elbow, on the other hand, is made largely of calcium and and support much larger internal stress while maintaining low enough strain that the tissues don't move too much relative to one another; e.g. the elbow doesn't break. Once the collision is over the bone molecules move back to where they were before. Of course, if the stress in the bone is too large, and consequently the strain exceeds a certain amount, the bone fractures.$^{[a]}$

You can think of the difference between pushing on a nose or an elbow in terms of pushing on springs with different spring constant. Supposing we have $F = k x$, then for a given force (stress) the displacement (strain) is $x = F/k$. A low $k$ means a large strain (like the nose) while a large $k$ means less strain (like the elbow).

Of course, there are also biological factors (but which are fundamentally physical of course). Certain parts of the body are simply more important than others. An elbow-skull collision does not have equal damaging effects to the owners of said elbow and skull. The impulse imparted to the elbow causes compression of the bone which transduces the force to articulating structures such as the shoulder. The skull, on the other hand, transduces the impulse to e.g. the brain. Rattling a shoulder around may hurt, but rattling a brain around leads to unconsciousness.

$[a]$: There is fascinating data regarding the stress/strain curves for bone. At low stress the bone is essentially elastic. Past a threshold, the strain is a much steeper function of stress. Then at a critical point the bone fractures.

P.S. I've left out a discussion of exactly how/why body tissues are destroyed by excessive strain. In other words, why doesn't your nose just always spring back to its original shape after being deflected by an elbow? A careful description of this process on the microscopic level would make an interesting subject for another question.

• The "funny bone" is a good example of this most people have experienced first hand. You can barely bump something that's moderately hard. If you bump it an inch in either direction, you barely notice the hit. But right on that nerve, and your whole arm lights up like you've been electrocuted. Sep 28 '15 at 9:13
• There's also the matter of how the impact is felt. When you kick or punch someone, you're not only expecting it, but the force is being delivered in a controlled way, and the inertia remains within the leg. But when you are kicked, say in the knee, the sudden jolt of inertia can cause things to pop out of alignment, and not being prepared for it, the sudden motion and reaction to it can cause even more damage. Sep 28 '15 at 16:55
• It also matters where the momentum ends up. Many strikes are not intended to hurt, but to throw the opponent out of balance by transferring more momentum than the opponent can pass on to the ground in that direction. Sep 28 '15 at 20:18
• I'm definitely no expert on martial arts, but it seems to me that at least part of the answer should involve balance — if you knock me sufficiently off balance, not only will I feel the force of your blow on me, but shortly afterwards I might be feeling the force of the ground rushing up to greet me. Even if I manage to refrain from falling, knocking me off balance could then feed back into the ability to hit strategic target points. Sep 28 '15 at 21:05

This is a standard question on how Newton's third law applies when two bodies attached on the ground collide. It is true that whenever you hit your opponent they apply, by reaction, the same force on your body, but do not forget that the total resultant of the forces also takes into account the reaction of the muscles and of the ground. Namely, given two bodies, $1$ and $2$, we have $$m_1\mathbf{a}_1 = \mathbf{R}_1 = \mathbf{F}_{2\to 1} + \mathbf{F}_{\textrm{ground}\to 1}+\mathbf{F}_{\textrm{muscles internal reactions}}$$ and likewise for $2$. Although it is true that $\mathbf{F}_{2\to 1} = - \mathbf{F}_{1\to 2}$ there are still other components coming into play to calculate the overall acceleration, and those other ones depend on your interaction with the ground and your internal muscles reactions. The muscles reactions depend very much on the individual parts of the body that collide and some other factors like timeframe of the impulse and so on and so forth. All overall the bottom line is that, besides the action-reaction force (which is equal and opposite in sign), there are other contributions that are to be addressed to the single bodies only and their interaction with the ground and internal structure, and those play a role as well in the complete equation.

• Good point about transduction into the ground, but doesn't that just mean that the fighter who's more grounded has higher internal stress than the player who is less grounded (the less grounded one will be accelerated by the blow more than the grounded one)? Sep 28 '15 at 8:04
• Yes, but not so easily. The interaction with the ground takes into account how your muscles react to the external solicitations and how they translate this into overall acceleration to each part of your body (and this depends on how they handles the stress). Sep 28 '15 at 8:29