Accelerating Through Impact: Mandate or Myth
-- January 7, 2012
Every golf instructor I've ever read or
heard says you must
accelerate through impact.
Then they extend this advice to mean that you will hit the ball farther
-- more ball speed -- because you are accelerating the clubhead through the ball,
than if you were simply coming into the ball at a constant clubhead
speed. The first is good instruction. The second is not good
In January 2011, I visited Jim McLean at his headquarters school at
Doral (Miami, FL), and had a wide-ranging discussion with him and his
staff about physics and golf. One of the more interesting topics was
the question of clubhead acceleration at impact. It "seems obvious" to
everybody who considers the question that an accelerating clubhead will
hit the ball farther than a clubhead that is not accelerating. I have
never heard or read any instruction that doubts the fundamental truth
of this statement. But is it true?
In this article, we are going to consider the questions:
How much farther -- if at all -- will
the ball go?
|If we are going to
seriously consider the question, we have to state
the premise less vaguely. We are going to compare:
The question is: how much faster
(if at all faster) is case 2?
- Ball speed for a clubhead arriving at the ball at V miles per hour
with no acceleration.
- Ball speed for a clubhead arriving at the ball at V miles per hour
(same clubhead speed) with a positive acceleration of A.
Note that I did not ask, "Is case 2 faster?" I asked for numbers. Numbers matter!
If it is only a fraction of a mile-per-hour faster then, for all
practical purposes, there is no difference. It
makes no sense to say, "You get more distance by accelerating through
the ball," if that extra distance is so small you can't even measure it
reliably. However, if it is several miles-per-hour
faster, that is a distance gain worth seeking.
(Brief digression on this point. Much of my writing on this site is
aimed at custom clubfitters and clubmakers. I have seen so many clubmakers
spend a lot of time optimizing one
parameter while ignoring or giving short shrift to a different
parameter -- one that matters a lot more. Why? Because they know how to measure and
adjust -- sometimes with great precision -- the one they spend time on.
Old proverb: "When all you have is a hammer, you tend to treat everything as a nail."
This article is about instruction, not clubmaking, but we are
investigating a similar phenomenon -- focusing on something that makes
a tiny difference, perhaps at the expense of something else that is
Simplified analysis #1 - Forces on the ball
Let's use 100mph as the clubhead speed. I have done some computer
simulations of good swings, and you might see an acceleration of almost
500 feet per second per second (500 ft/s/s) just before impact. We'll
use 500 ft/s/s as the clubhead acceleration for case 2.
Impact lasts 0.0004 second (0.4msec). During that time, the force the
exerts on the ball averages about 1600 pounds. (We need that
much force in order to accelerate a 46-gram ball to 150mph in 0.4msec.)
For this analysis, we focus on the forces between the clubhead and the
ball. The clubhead is a low-loft driver, so we don't have to worry
about a glancing impact; we can use the simple impact formulas.
If the clubhead is not accelerating at impact,
that 1600 pounds is the force. Period!
If the clubhead is accelerating,
then there is a force on the clubhead making it accelerate. F=ma! We
decided to use 500 ft/s/s as the acceleration at impact. When we crank
this into F=ma (along with 200g
for the mass of the clubhead), we get a force of about 7 pounds
accelerating the clubhead.
Let's make the most generous assumption: all the force of the
clubhead's momentum and all its accelerating force helps to accelerate
ball. So we have:
So the total force accelerating the ball is 7+1600=1607 pounds. How
much larger is that than the force in case 1? The increase is 7/1600 =
.004 of the total force = .4%
pounds due to the clubhead's momentum.
- 7 pounds
accelerating the clubhead.
That is less than half a percent. So the acceleration is worth less
than an additional
1/2mph of clubhead speed, or an extra distance of less than 1.5 yards. Hardly worth concerning yourself with,
unless you are a long drive competitor and pure yards are the way you
Simplified analysis #2 - Clubhead speed
Let's try another way of looking at it. What would the clubhead speed
be immediately after impact,
if the clubhead had not lost any momentum from impacting the ball?
Why would this matter? If the ball were not there, we have a speed
before impact and another one 0.4msec after
impact (i.e.- after separation). The clubhead speed that would be
applied to the ball is somewhere
between these two. So let's find both.
The clubhead speed coming into impact is 100mph. That is one of our
Since Case 1 is the case of zero acceleration, the speed 0.4msec later
is still 100mph.
Again, by assumption, the clubhead speed coming into impact is 100mph.
We have an acceleration of 500 ft/s/s. Therefore, in the duration of
impact, the change of velocity is given by the acceleration times the
time the acceleration is happening:
Change of velocity = 0.0004sec * 500 ft/s/s = 0.2 ft/sec =
Velocity 0.4msec after impact = 100 + 0.14 = 100.14mph.
So, out of a clubhead speed of 100mph, the acceleration gives us a
boost of only 0.14mph. That's only than a 0.14% gain, and represents an
extra 15 inches of distance on the drive. Note that it is
upper bound of the gain in
ball speed that you could expect; the actual distance gain will be
less. It is even
less than Simplified Analysis #1; this one is probably negligible even
for a long drive competitor. (Not zero, true. But negligible.)
Our two back-of-the-envelope calculations show distance gains of no
more than 4 feet and 1.3 feet respectively, and probably less than that.
That's feet, not yards. In the appendix, I have a more
that comes closer to
an exact answer -- and it is even lower than the two simplified
analyses. It only shows 8
of additional distance.
For comparison, let's look
at just a few other things you might be working on instead of
accelerating the clubhead:
So physics does not back up the notion that accelerating through impact
will increase distance. At least not by any amount that you would
notice, or would do you any good. Certainly there are better things to
worry about, if driving distance were the only benefit of accelerating
- Improve your wrist flexibility.
Just one degree more of wrist cock is worth a yard and a half of extra
distance. (No, not a lot. But almost seven times what clubhead
acceleration is worth.)
- Get contact on the sweet spot.
With most drivers, missing the sweet spot a half-inch low on the face
will cost ten yards. (From my article on gear
effect, and the HotStix data cited.)
- Work on your core strength.
An improvement of just 2% in the strength of your body rotation is
worth two and a half yards.
Which brings us to our second question...
Is it good instruction?
We just found that it
makes no sense to say, "You get more distance by accelerating the
the ball," if that extra distance is so small you can't even measure it
reliably. So why even mention it in instruction? Actually, I am in
teaching accelerating through the ball. But not because clubhead
acceleration at impact will hit the ball farther. There are quite a few
things that are bad physics but good instruction, and this is one of
them. Let's look at some reasons that accelerating through impact is a
good swing key.
WHAT IS REALLY ACCELERATING?
Is it the clubhead, as is usually assumed by teachers telling us to
"accelerate through impact"? If not, is it the hands? The body? Let's
look at the possibilities.
already debunked the notion that you get more ball speed from an
accelerating clubhead. But there's another reason not to try to
accelerate the clubhead -- at least not unless you are doing it right.
The act of trying to add clubhead acceleration may itself be deleterious to
clubhead speed, if you do it in the obvious way. That seems like a non
sequitur -- it doesn't follow. But let's review where clubhead
acceleration can come from. We're doing physics here.
The picture at the right shows the two ways the clubhead can still be
accelerating at impact. One is good and the other not so good.
Let's start out with the not-so-good one, the upper example. The wrists
can be applying a torque to the handle, creating an accelerating force
on the clubhead. The shaft must bend backwards to transmit the force.
The reasoning is presented in my
article on the right-hand hit.
Most of that same article examines whether this sort of acceleration
provides more clubhead speed or less. The conclusion is that, if
applied with exquisite timing, strength, and hand speed, it might give
a tiny bit more clubhead speed. But miss out on the timing, strength,
or hand speed and it will cost far more distance than could possibly be
gained by perfect application.
So you don't want to accelerate the clubhead directly -- by forcing it
with the hands and wrists.
The lower example is a more productive way to apply clubhead
acceleration through impact. The picture (which I adapted from Andrew
Rice's web site)
shows the proper position at impact for an iron shot, as exemplified by
an impact bag drill. You want the hands leading the clubhead. This
implies that the wrist cock is not completely released at impact; the
green angle (which I added to Andrew's photo) shows the wrist angle at
If you understand how
centrifugal release works
("inertial release", if you're going to be a stickler about
terminology), you will recognize that this position has the clubhead
still accelerating; the inertial release has not yet completed. I said
this is more productive than accelerating the clubhead with wrist and
hands. But why? There are a number of reasons.
So there are good things that happen if you retain a bit of wrist cock
at impact, especially with irons. One of the things that happens is
that the clubhead is accelerating at impact -- but that is just a side
effect, and doesn't really help you.
- Go back to the examples
in my article, particularly the "strobe" pictures drawn by the SwingPerfect
You will note that using your hands and wrists to retain the lag (as
opposed to forcefully helping release the lag) will actually increase
clubhead speed. Also from the examples, note that a side effect is that
the wrist cock is not completely released at impact. So -- lo and
behold -- you do increase ball speed.
But not because you are accelerating the clubhead at impact; it is
because you held off centrifugal acceleration until late in the
downswing, where it creates the highest clubhead speed.
an iron shot, you want the hands to lead the clubhead
for purposes of a solid strike and a negative angle of attack. This
becomes less of an advantage with fairway woods and drivers; the usual
advice there is to "sweep" the ball rather than hitting down on it. But
even with woods, you don't want to go beyond a flat wrist at impact. A
cupped wrist is a bad idea; the same
physical argument says that the clubhead is decelerating if you do
that. Which, of course, means that you missed the maximum clubhead
speed, which occurred before the clubhead reached the ball.
|The body turn involves angular
velocity and acceleration, so "accelerate through impact" might refer
to body rotation. In this category, I will lump together legs, hips,
trunk -- all the way up to the shoulders. That's because all have to be
involved to move the shoulders. The muscles of every link in that chain
must either create motion or trasmit it, up from the ground to the
What happens if the body does not accelerate through impact? Well, once
the ball is gone, anything that happens makes no difference. So,
theoretically, acceleration beyond impact is not going to matter. But
that's theory, and someone once pointed out to me, "The difference
between theory and practice is bigger in practice than in theory."
The problem with this theory is the difference between intent (or even
"feel") and what the body is actually doing.
This relates very directly to accelerating the body turn through
impact. A real live human being cannot accelerate the body to impact
and not still be accelerating through impact.
A golfer thinking only of accelerating to impact is going to quit
accelerating before getting near impact. You can't turn off full-body
acceleration in .4 milliseconds, probably not even in 100
Here, then, is the situation. If you want to maintain the body's
angular acceleration up to impact, you must intend
to accelerate through impact.
If there is any reason to want angular acceleration to continue fully
to impact (and we will see below that there are good reasons), then a
good instructor will teach the golfer the intent of accelerating
In fact, a good instructor will have the golfer exaggerate...
Accelerate well beyond impact. If you have taken a lesson where your
swing was video'd, and watched
the video afterwards, you know this is true. You may intend to make a
particular exaggerated move, you may have felt
that you made the move successfully, but it is barely there (if at all)
on the video. Much of good golf instruction is getting people to
greatly exaggerate a correct move, because it is the only way to get
the motion to happen at all. So, assuming we want angular body
acceleration to continue up to impact, the swing key to be taught is to
accelerate well beyond impact.
|I subscribe to golf instruction
that says that the hands are moved by the body. But that is not the
only theory of instruction out there; I have read books that say things
like, "The arms do the swinging part of the golf swing... The body does
not swing. It reacts to the swing." I may disagree with that, but it might be a productive intent and feel for some golfers.
And that makes it valid instruction for those golfers.
But -- make no mistake about this -- physics says that hand and arm
motion is caused by body rotation. That is actual, as opposed to intent
and feel. So, if we are going to analyze the physics of the swing, the
motion of the hands and arms is driven by the body rotation. For the
first approximation, we do not have to analyze hands and arms
separately, just the body rotation -- which we discussed above. (If we were to refine the analysis, which I won't here, we
would next account for the left arm's separation from the
body late in the downswing. Still nowhere near "arms motivating the
swing", but at least there may be some change of the result due
to the rotation of the arm not being exactly the same as the rotation
of the body.)
CONSEQUENCES OF ANGULAR DECELERATION
Above we discussed the angular acceleration of the body. The argument
above maintains that, unless you try to accelerate the body's rotation
well through impact, you will in fact lose acceleration well before
impact. In golf terminology, you will "quit on the shot." This has a number of
To demonstrate the points above, I ran more swing simulations with
SwingPerfect. The way you maintain angular acceleration is to apply
torque to the body to twist it. With a proper swing, this twists the
entire assembly of body, shoulders, arms, and hands. SwingPerfect
allows you to adjust the body torque over the course of the downswing
-- not a very fine adjustment, but enough to see the effect of
modifying the torque-vs-time curve. I ran three different swings for
comparison. Here are the results.
actually is lost. This has nothing to do with the clubhead
accelerating through the ball, but rather that less acceleration is
applied to the clubhead for tens of milliseconds before
impact. The clubhead is deprived of some of the accleration it would
have had, in order to build up speed at impact. And clubhead speed at impact is what really matters for distance.
left wrist is cupped, rather than the proper flat or even bowed
left wrist. We know that a bowed wrist at impact (or a flat wrist at
the very least) is desirable for a solid hit. A cupped wrist, on the
other hand, is associated with lost clubhead speed, fat or thin shots,
or too-high balloon shots. It is also likely to point the clubface to
the left, resulting in a pull or even a pull-hook. (Think about it this
way: if the swing plane were perfectly vertical, then a cupped left
wrist would add loft. Since the swing plane is not vertical but tilted,
some of that "loft" turns into a left-facing angle.)
Torque variation during downswing
|Constant accelerating torque
the downswing, no "quit".
(For later reference, this is 60*280=16,800 foot-pound-milliseconds.)
- The torque is 60 foot-pounds.
|Our best approximation to the
classic "quit" scenario:
- The same 60 foot-pound of torque for the first 200msec of
- Then the torque drops to 40 foot-pounds for the last 80
|Suppose, even with "quit", we
apply the same total torque-times-milliseconds to the swing as we did
earlier to the constant-torque swing. Real
swings don't work like that, but let's see how much of the effect is
due to the total torque-seconds applied and how much to the fact that
torque is withdrawn late.
(66*200 + 46*80 = 16,880 foot-pound-milliseconds, almost exactly the
same as the constant-torque swing.)
- A torque of 66 foot-pounds for the first 200msec.
- A torque of 46 foot-pounds for the last 80 msec.
This table confirms the consequences we guessed at above:
Let's take one last look at the wrist cupping when the body turn
quits coming into impact. Every instructor knows this is true! But the
average golfer, and even the average instructor, might not understand
why it must be true. It's not complex muscle interaction, it's just
physics and geometry. I set up a simulation with exaggerated "quit" to
illustrate it, using SwingPerfect. Look at the "strobe" swing
- If you deprive the downswing of body torque by quitting on it,
you lose clubhead speed -- and, obviously, distance. Not only that: the
torque late in the downswing (the torque you lose when you quit) is
more important than torque early in the downswing. How do we know?
Because we still lost 3mph of clubhead speed (about 9 yards) in the
third calculation, where the total torque-seconds were the same as the no-quit swing; the
torque was larger early in the downswing and reduced during the "quit"
at the end.
- Loss of body torque late in the downswing will indeed turn a
bowed wrist (good!) into a cupped wrist (bad!).
- The first animation maintains body torque through impact -- that
is, it keeps accelerating the body rotation through impact. The
clubhead starts with a lot of lag, and really gets moving (develops a
lot of momentum) late in the downswing. The clubhead has just about
caught up with the hands as it impacts the ball, giving a flat or
slightly bowed wrist position at this position.
- The second animation starts out the same, and maintains
acceleration for the first 200msec of the downswing. Then it gets
seriously torque-deprived. That is what we mean by "quitting on the
swing"; we don't apply energetic body turn through the ball. Because
we're only human, the loss of acceleration takes effect before we get
to the ball -- earlier than the quitter intended. You can see that the
hands (the red dots) are slowing down late in the sequence; the red
dots are closer together, so we know they are getting slower.
Meanwhile, the clubhead (the gray dots) has attained quite a bit of
momentum, even after just 200msec. True, not as much as if we had
continued to accelerate the rotation, but enough to catch and pass the
hands as they slow down. At this point, it's just geometry. The last
tens of milliseconds before impact, the slowing hands and not-slowing
clubhead result in a cupped wrist -- and all the evils that the cupped
SHORT GAME IMPLICATIONS
So far, we have been talking about a full swing. But there are two
pretty obvious implications for the short game as well. I haven't
worked the numbers on this, but logic and plenty of anecdotal evidence
seems to confirm it.
- In putting, the body turn becomes shoulder rock; the swing plane is
close to vertical. If you don't continue the shoulder rock well past
impact, your wrist will cup. That tends to cause a pull. People who are
"wristy" putters have learned to time the release to compensate for it.
But it is hard to teach, and a low-percentage high-maintenance stroke. Now that we know this little gem of information, putting
instruction has adopted a back-and-through motion with as steady a left
wrist as possible.
- For these shots, wrist cock is negligible to nonexistent, so
centrifugal acceleration isn't going to be a factor. What is left is F=ma;
accelerating the clubhead through impact means that the hands, through
the shaft, are applying a force to the clubhead. With a clean strike,
the accelerating clubhead doesn't matter. But a slightly fat hit will
take more than 50% off the distance of a chip or pitch, as opposed to
maybe 10% or 20% from a full swing. That is because the clubhead is not
just transferring momentum to the ball; it is losing momentum to the
ground before it even gets to the ball. When the clubhead strikes a
ball, the only resistance is inertia. But the ground is adhered to the
earth, and the earth is for all practical purposes an infinite mass. So the club has to
administer enough force to cut some turf away from the earth. This is a
lot more force than just moving a free-sitting golf ball. So a lot of
the clubhead's momentum is lost providing the force needed to cut the turf.
does this have to do with the accelerating force? Well, force applied
to the clubhead can restore some of the lost momentum due to the fat
hit. If it is seriously fat, then not much will be restored. But those
slightly fat chips will get closer to where you want them if you are
accelerating the clubhead all the way through the ball.
Still, like putting or even the full swing, you don't want the hands to create the acceleration, just transmit
it from the shoulders. "Wristy" chips and even pitches are
lower-percentage shots. Unless they are perfectly timed, the usual
problems of a cupped wrist are likely to occur: fat hits and sculls.
(Think in terms of Steve Stricker's motion; he is one of the most
reliable wedge players in golf. He sets the wrist angle, then uses his
hands, wrists, and forearms to keep the angle. The acceleration for the
swing comes from the shoulders, and continues well through the ball.)
"Accelerate though the ball" is an excellent swing key, for every shot from a drive to a putt.
But it is better expressed as "rotate your body through the ball"
(or perhaps "accelerate moving your hands beyond the ball"), because it
is body rotation or "turning the triangle" where acceleration through
the ball is beneficial. Consciously accelerating the clubhead through
the ball (especially if you do it with the hands, wrists, or forearms)
will probably do more harm than good.
First off, I'd like to thank Jim
McLean for asking the questions that sharpened the physics
issue, to the point that there was something worth writing.
Then Bob Corbo of Simductive
Golf got on my case about wrist position and acceleration at
insisted that he and all his students had much more solid ball-striking
and never an over-the-top if the wrist is flat or bowed from the
transition through well past impact. He challenged me to explain why.
It was clear it had something to do with acceleration.
Appendix - More exact analysis
Here is a more detailed analysis of the physics of impact. We are going
to look at the equations for momentum transfer from clubhead to ball,
and see how that is affected by an accelerating clubhead. If you're not
into the math and physics of it, you can skip the details below. Here's
a peek at the ending: The distance gain you can expect from an
accelerating clubhead is less than a foot -- only 8 inches.
As in the simplified analyses, we will compare:
clubhead impacts ball at constant speed of 100mph.
clubhead impacts ball at speed of 100mph, while accelerating at
These two cases will be compared, to test the value of clubhead
acceleration in creating ball speed.
We'll start with notation. The computation will be done using MKS units
(meter-kilogram-second), to be sure we aren't making mistakes in units.
We will convert back to the units we recognize (like pounds, miles per
hour, etc) when we want to get a feel for what's going on.
- U denotes a velocity just before impact.
- V denotes a velocity just after impact.
|= Clubhead speed just
before impact = 100mph = 44.7 m/s
Ball speed just before impact = 0
Clubhead speed just after impact (we will compute)
Ball speed just after impact (we will compute)
Clubhead mass = 200g = .2 Kg
Ball mass = 46g = .046 Kg
|= .046/.2 = 0.23
(Listed here because we use this ratio a lot)
Coefficient of restitution = .83 (max allowed
CASE 1 - NO ACCELERATION OF CLUBHEAD
This is a simple momentum transfer problem.
The clubhead impacts the
ball, and transfers some of its momentum to the ball. Since the
clubhead is more than four times as heavy as the ball, it will still
retain some considerable speed after impact -- though a lot less than
before impact. The total momentum of clubhead plus ball remains the
same after as it was before. And, of course, the total energy after
impact will be less, because the COR is less than 1. Some energy will
be lost to internal friction during the collision (mostly friction
inside the ball, but a little in the clubface).
We will start with the well-known
equations for a simple lossy impact.
|C m (Ub-Uh)
+ M Uh + m Ub
|C M (Uh-Ub)
+ M Uh + m Ub
But we know the ball starts at rest (Ub = 0), so we can
|M Uh - C m Uh
1 + m/M
|C M Uh + M Uh
1 + m/M
That last equation should look familiar; we
use it all the time to
compute ball speed.
The first equation is something we don't see as often; it gives the
clubhead velocity after impact. It will be slower, because the clubhead
has transferred a bunch of momentum to the ball. If we use numerical
values we originally assigned for m, M, and C, the equations
- (.83 * .23)
1 + .23
Uh * .66
1 + .23
Uh * 1.49
Again, the second equation is familiar; it
is the maximum legal "smash factor" for a 200g driver head. It is
generally thought of as 1.5, which is a very close rounding of 1.49.
The first says that the clubhead loses about a third of its speed in
Now it's easy to solve for Vb and Vh, the ball and clubhead speeds
= 149 mph =
= 66 mph =
So now we know how much the head slows
down, and the ball speeds up, during
here, let's look at the force and momentum involved.
Momentum is interesting to physicists at
least in part because it
relates force, time, velocity, and mass. Here's how:
Everybody know's Newton's
second law: F = ma
One component of that is acceleration, which is a change of velocity
over time. Physicists and
mathematicians usually refer to such a change as a "delta". So a change
of velocity would be "delta V" or ΔV. So:
Ft = m ΔV
The right side is the change of momentum.
The left side is called
"impulse", and the relationship is: Impulse
equals momentum change.
Physicists and mathematicians have a more general view of impulse; it
does not require a constant force, as our equation does. But we prove
below that, for our calculations, a constant force gives exactly the
same answer as if we plotted the exact buildup and decay of force
during the collision between clubhead and ball.
Let's use this, plus our results above, to
compute first the momentum
change for clubhead and ball, then the force the clubhead exerts on the
ball to cause the momentum change:
mΔV = M (Uh - Vh) = .2
Ball: mΔV = m Vb = .046 * 66.6 =
Good, they're the same! That means we did
arithmetic right, because the law of conservation of momentum says: any momentum gained by the ball must be lost
by the clubhead. And it was.
Now for the forces. From above, impulse =
Ft = mΔV
We know that the momentum change is 3.0
change = Impulse = Ft = 3.0
we also know that t = .0004, so:
F = 3.0
F = 7500 Newtons (N) = 1686
So the average force between clubhead and ball during impact is 1686
CASE 2: CLUBHEAD ACCELERATES INTO BALL
In a good swing, the clubhead can
accelerate into the ball by as much
as 500 feet per second per second. (In
MKS units, that's 152.4 meters per second per second.)
The only reason clubhead
acceleration might matter for distance is that
there is an extra force added to the momentum transfer that we had in
case 1. That force is whatever force is accelerating the clubhead. If
it was accelerating the clubhead coming into impact, then it is
continuing to push the clubhead during impact -- and that is where any
benefit will derive.
Let's use F=ma to see how
much force is
accelerating the clubhead just
F = ma = .2 * 152.4 = 30.5 N
This force will increase the momentum of
the whole system during
impact -- both the clubhead and the ball it is "pushing". For the .0004
seconds that the clubhead and ball are in
contact, we will have an extra 7 pounds of force (we'll round 6.85 up)
accelerating the whole
246-gram mass of clubhead+ball. Let's see what that does to the final
velocities. (The calculations below are in MKS units.)
F t = (M + m) ΔV
M + m
.200 + .046
.0496 m/s = 0.11mph
So it has added a whole .11 mph to the
combination of clubhead and ball
by the time the ball separates from the clubhead. That means:
speed after impact
66 + .11
speed after impact
149 + .11
This is not much of a gain in yardage.
Using the rule of thumb
'1 additional mph of ball speed gives
2 additional yards', we have
increased the distance by 0.22 yards, or about 8 inches.
Why is this less than the rough
calculations, the simplified analyses, in the main body of the article?
Because each simplified analysis was a 'back of the envelope'
calculation that I
could do in a couple of minutes. Any time you make a rough calculation
instead of a complete analysis, you must make lots of
assumptions. In each case, I had chosen an assumption to make
acceleration look important. I did that because I suspected the answer
would come out, "It's not that important." I didn't want my assumptions
challenged as the reason it didn't look important, so the assumptions
were all in favor of acceleration. Note that, even so, acceleration
didn't seem all that important. But when I replaced assumptions with
actual calculation, the results showed acceleration to be even less
DEPENDENCE ON FORCE PROFILE DURING
point: These calculations are
based on a constant force -- the
average force -- during contact between clubhead and ball. The blue force-vs-time
profile in the picture shows
this; during impact, the force is some constant value, and it is zero
all other times. Obviously
this is contrary to fact; the force starts small, increases as the ball
compresses, and decreases as the ball releases and leaves the clubface.
In other words, the red force-vs-time
profile is what is really happening.
So we need to show that it doesn't matter; any
force profile that gives the same
will give the same final velocities. ("The same average force" means
the same area under the profile curve.)
Below is a quick proof, in
you're interested. It involves a bit of integral calculus, but very
The only place we make assumptions about the force profile is where we
use 'impulse equals momentum change'. So let's generalize that, using
force as a function of time.
That means we re-express F t = M ΔV
dt = M dV where d means an
To find the actual change in velocity , we need to integrate.
F(t) dt = M ΔV
or, solving for ΔV:
ΔV = 1/M * ∫
Equation #1 gives the change in velocity if we used the actual
force profile F(t):
start small, increase to a peak, then fall off
again. In order to compute ΔV directly from physical principles, we would need the
actual force function
then use integration to find the area under the curve.
But, in our calculations above, we used the average force over the
of impact, Fav. Is this legitimate? Does it give the same answer?
To find out, we have to see how the average force Fav
is computed? If the total time is T (in our case,
0.4 milliseconds), then the average is computed as:
Fav = 1/T * ∫
In our presumably "exact" analysis, we used Fav to compute ΔV using the formula:
ΔV = Fav * T
Let's plug in for Fav the time-dependent formula for Fav using integration. So ΔV becomes:
ΔV = 1/T * ∫ F(t)dt *
T / M
ΔV = 1/M ∫
That's exactly the same as Equation #1 above. Which says we get the
same change in ball speed whether we use the average force or the
actual force profile.
modified -- January 13, 2012