Martial arts moves can seem magical, but maybe they just display a mastery of physics.

Photograph: Stanley Bielecki/Getty Images

Martial arts have a sort of
magical aspect. It can seem like those who have mastered them have
ventured beyond the realm of physical possibilities and obtained a
superpower. In this case, I'm going to examine the â€œ1-inch punch,â€ which
Bruce Lee made famous at a 1964 karate tournament, delivering a
powerful blow with his fist starting just an inch away from his
opponent.

This
punch seems like it should be impossible. I mean, if a mere mortal were
to punch someone, they would pull their fist back a good distance
before striking. Punching over such a short distance is like jumping
really high without bending down first. Let's figure out what's going
on.

Forces and Momentum

I'll
be honest: This is an excuse to talk about some of my favorite physics
concepts, force and momentum. If two objects interact in some way, such
as by pushing on each other, then we can model this interaction as a
force. (You must have at least two objects to have an interaction.) When
object A pushes on object B, B pushes back on A with a force of the
same strength.

Here's what that would look like as a physics diagram:

Courtesy of Rhett Allain

It's important to remember that force is a property of the interaction, not a property of the object.

The
force on an object changes its momentum, a measure that is the product
of an objectâ€™s mass and its velocity. (Stationary objects have a
momentum of zero.) If more than one force acts on an object from more
than one interaction, then the totalâ€”or netâ€”force changes the objectâ€™s
momentum.

Before
we get to the punching, there's one more important thing to consider in
our mini physics course, and it has to do with the nature of â€œobjects.â€
In short, stuff is made of other stuff. If you want, you can model a
tennis ball as a single objectâ€”but it's not really a single object. In
fact, a tennis ball is made of many parts, and each of these parts are
made of molecules, and each of those molecules are made of atoms. If you
have a single force acting on a tennis ball, it is actually creating a
vast number of interactions between an uncountable number of atoms.

No one wants to deal with that many interactions.
Instead, in physics we treat the ball as one thingâ€”and that's mostly
fine. However, to make sure that other people understand what we are
doing when we model an interaction, we have to define our â€œsystem.â€
Perhaps, to make it easy, we decide the system is just the ball itself.
If so, we deal only with the ball's momentum and any forces due to
external interactions, and we can ignore all those atom-atom
interactions. We could even forget about the interaction between the
ballâ€™s fuzzy surface and its interior rubber part.

It's
also possible to have a system that consists of more than one object.
Imagine a tennis ball attached by a string to a soccer ball. If I want
to use a system consisting of both balls, then I would only look at
forces due to external interactions. I wouldn't include the force the
string exerts on either ball.

For the momentum of
this system, I would use its total mass, which is the sum of the mass of
the balls, and the velocity of the center of the systemâ€™s mass. Since
the soccer ball has a larger mass, this center of mass would be closer
to it along the string and farther from the tennis ball.

Courtesy of Rhett Allain

Guess what? Humans are also made of stuff, and a
person also has a center of mass. But the physics of humans can get
messy, since they can change shape. Different parts, like arms and legs,
can be positioned differently. However, a good rough estimation is that
the center of mass for a standing person is somewhere between their
belly button and their spine. For a person in a sitting position, their
bent legs will move their center of mass a little closer to their chest.

The System of Bruce Lee Plus the Target

From
a physics perspective, any punch can be complicated. So let's make it
as simple as possible by considering the 1-inch punch for a system
consisting of one puncher and one punchee. Letâ€™s call them Bruce and
Joe, respectively, since there's a famous video of Bruce Lee punching martial artist Joe Lewis at an exhibition.

With
this system we can ignore any forces due to internal interactions. Yes,
that means that we don't actually have to look at the force from the
1-inch punch. It's an interaction between two objects in the same system
(Bruce and Joe).

What forces do we have left? Really, there are just
two external interactions. There's the downward-pulling gravitational
force from their interaction with the Earth, and there's the interaction
between the floor and the system. This floor force can push up and also
sideways, because of friction.

What
about the systemâ€™s center of mass? We need to know something about the
positions of Bruce and Joe. Usually, both people start standing up, and
the puncher positions their fist 1 inch from the target. After the
punch, the punchee falls back into a chair thatâ€™s been handily
positioned behind them.

I'm going to draw a
stick-figure version of this action both before and after the punch,
along with the approximate center of mass represented by a red dot.

Courtesy of Rhett Allain

Let's look at the motion of this center of mass for
the system of Bruce plus Joe. First, you can see that the center of mass
moves to the right. It's still in between Bruce and Joe, but since Joe
moved to the right, so did the center of mass.

Next, you should notice that the height of the center of mass moved down.
Why? Well, Joe fell into a chair. That means that Joeâ€™s center moved
down, which decreased the overall height of the system (Bruce plus Joe).

Finally,
the center of mass has a velocity moving to the right. Just after the
punch, Joe's still sliding in the chair, so his location is also moving.

How
can we explain this motion of the center of mass just from the external
forces? Of course, the gravitational force pulling down on the system
can account for the downward motion of the center of mass. And thereâ€™s
the upward pushing force from the floorâ€”but really, that just prevents
the system from falling down below the level of the floor. Then what
force makes the center of mass move to the right and increase in speed?

The answer is friction. When Bruce does his 1-inch
punch, there's a frictional force between the floor and his feet pushing
to the right. This frictional force pushes the center of mass to the
right.

What if Bruce did his famous punch while
standing on ice? There wouldn't be an external force from friction. Yes,
Joe would still move to the right from the punchâ€”but Bruce would recoil
and move to the left such that the center of mass would be horizontally
stationary. (It would still move downwards, because Joe fell.)

The System of Just Joe Lewis

You
might think it's silly to look at the system of both humans, but it
shows us that the frictional force is quite important in the overall
result. But what if we look at the system of only Joe Lewis?
From the motion of Joe's center of mass, we can get some idea about the
forces acting on him. Yes, one of these external forces pushing on Joe
is Bruce Lee's 1-inch punch.

Let's get some real data on Joe's recoil. I'm using a clip from this compilation of punches,
and I'm just guessing that the part in black-and-white film is the
punch with Joe Lewis. If not, that's coolâ€”it really doesn't matter which
person plays the target, since they don't have an active role. Now, I'm
going to use Tracker Video Analysis) to mark Joeâ€™s location in each frame. From this I get the following for the horizontal position as a function of time:

Courtesy of Rhett Allain

After the punch, his horizontal position changes at a fairly constant
rate, so that the slope of this line can give his horizontal velocity.
From the analysis, this puts his velocity at 1.19 meters per second.
With a mass of 70 kilograms (which is just a guess), that means he has a
change in momentum of 83.3 kilogram-meters per second. (Kg*m/s is the
unit for momentum.)

This number is very useful. Since this change in
momentum is related to the force that was exerted on him from Bruceâ€™s
punch, we can write this as the following expression:

But
we don't actually know the contact time. That's fine. Let's just make a
rough estimation from the video, which shows that Bruce's fist is in
contact with the target for about three frames. This particular clip is
running at 25 frames per second, so three frames would be 0.12 seconds.
This gives an average impact force of 694 newtons, or 156 pounds. That's
the amount of force it would take to lift a fully grown human (but only
for a very, very short time). I don't think this force value is
extraordinarily largeâ€”but I'm also not saying that I could do it.

Before
we move on to the Bruce Lee system, there's one more important thing
about this punch. It's sort of a trick to put the chair behind the
punchee. This makes the impact seem more dramatic than it actually is.
Let me draw the horizontal forces on Joe during the punchâ€™s impact, and
you can see how this trick works. (I left off the two vertical forces
from gravity pulling down and the floor pushing up.)

Courtesy of Rhett Allain

In the horizontal direction, there are just two forces: the force from the punch (F_{B}) pushing to the right and a weaker frictional force (F_{f})
pushing to the left. Since the net force pushes to the right, Joe will
increase in momentum to the right. But notice that the frictional force
is applied to his feet and the punch is somewhere around his chest.
Since these two forces are applied at different locations on the body,
they will cause a rotation about his center of mass. That means he will
tip over and fall. Good thing that chair is there waiting for him.

Of
course, standing up straight with your feet close together isn't such a
great idea. This knock-over wouldn't be as easy to accomplish if Joe
had his feet apart. With one foot back, the upward-pushing force from
the floor would counteract the rotation of the other two forces.

The System of Just Bruce Lee

This
is what you have been waiting forâ€”and why I put him last. I've already
estimated that Bruce Lee exerts a punch force of around 694 newtons.
Like I said, it's not the force that's impressive, it's the short
punching distance. Heâ€™s only punching 1 inch, which is 2.54 centimeters.

Let's
compare this to a punch from a more normal distance. Suppose Joe wants
to return the favor to Bruce. It seems safe to assume that Joe's punch
could also deliver a force of 694 newtons, or somewhere around that
value. However, this punch accelerates his fist over a distance 0.5
meters instead of 2.54 centimeters. (I estimated that distance by
pretending to punch someone and noting how far my fist had to move.)

Let's
calculate the force-to-distance ratio for these two punches. The ratio
for Joe would be 1,388 newtons per meter, but Bruce's would be 27,300.
That's almost 20 times greater. He must be a superhuman.

Oh,
just one second. There's something else going on. If you take a look at
Bruce's 1-inch punch very carefully, you will see something useful.
Bruce doesn't just move his fist forward 1 inch. Before the punch, he
actually moves his whole body forward. (He doesn't pick up his feet, but
he definitely moves his body.) If you were to track the location of his
center of mass, you would get the following plot of his horizontal
position as a function of time:

Courtesy of Rhett Allain

Notice that most of this center of mass motion is before
the punch. Looking at the slope of the best fit line, it seems like he
is moving about 0.36 meters per second in preparation for the punch.

Does
this even matter? Letâ€™s do another calculation. Let's say Bruce is
moving with this speed towards a stationary Joe and they collide, but
there's no punch. After the collision, Joe recoils with some velocity
and Bruce just stops. If the only interaction is due to the collision,
and Bruce and Joe have the same mass, then with Bruce stopping, Joe
would recoil with a speed of 0.36 m/s. (You can see this same thing
happen when two pool balls collide and one stops while the other travels
away at the same speed.) This gives Joe a lower fallback speed, but
it's not tiny either.

With Bruce moving his whole
body, itâ€™s almost like he has a second â€œfistâ€ thatâ€™s punching the
target. This second fist has momentum, even though itâ€™s not moving very
fast, because it has the mass of his whole body. Also, by moving his
whole body, Bruce can essentially increase the total time of the punch
without actually touching the punchee. It makes the 1-inch punch a whole
body interaction using his legs, instead of just his fist.

So
what can we now say about the physics of the 1-inch punch? First, if
you have the target standing with their feet close together, that person
is going to probably fall back, even if a mere mortal like myself
delivered the punch. Second, this isn't really a â€œ1-inch punchâ€ since
Bruce is actually moving his whole body over a larger distance.

I
think we can all agree that physics gets the credit here, not magic.
And so does training and skill: Bruce Lee could deliver a punch with
quite a significant wallop. In the end, it doesn't matter if this punch
is superhuman or notâ€”I don't want to be on the receiving end of it.