Don´t worry, Yurek, about your lazy holiday, I´m sure I can´t finish the task alone  . Autumn will be a good time to work on physics.  Thinking of different ways to measure the initial speed of the projectile, as you and Matthias are talking, I see that another way, less accurate than the ballistic pendulum (wich is exact if well made), could be one working on the penetration of the projectile into a soft target. It coul be a first aproach, simple to make for  anybody. The kynetic energy of the projectile will be fully spent making the work of penetration. This work is the product of the penetration length and the resistance force of the material. This force would be calculated for a projectile from the resistance pressure (Kg/cm^2, etc) of the material, wich is a data of the manufacturer. It could be also easily calculated loading weights on the material until overcoming its resistance. From the equivalence "penetration work = kynetic energy" we extrac the speed.

A cheap an easy to get material would be the expanded polyethylene, that is used for modern targets in archery, though a material of very fine grain would be better and would provide more accurate messurements. If the plasticine were enough cheap as to make a big target...  

Looks like you are on the right track Yurek. I certainly have the feeling while slinging that it is strength rather than speed that is limiting my range. Given a sling with little or no drag (there's that vacuum again) The amount of force you can put into the snap should be the limiting factor. I'd love to see a very high speed fram by frame analysis of a well executed throw - I think the forces/tensions achieved would surprise many of us...

I like the idea of projectile penetration too, but am hesitant based on too many years setting up experimental apparatus.

I think I "might" be able to swing a high speed strobe without spending any money. I'll build an LED enabled glans first. The unfortunate part is that with me as the only model, we won't be capturing the pinnacle of modern slinging by any means!

Matthias

Can you guys write up a little outline for what you will cover in the Physics section.  i.e.:

PHYSICS SECTION
- Mechanical Advantage (intro, why a sling works).
- Orbital/Rotational Physics (tangential releases, etc.)
- Drag
- Trajectories
- ....
- ....
etc etc.

I'm not a physics buff, so my outline isn't meant to be serious, but perhaps something like it.  I think will help you focus the section and help you divide up the topics among your selves (and have clean divisions).  I can also update the master outline with all the subtopics that will be included.

Chris

Agreed. I think we are still in the stages of figuring out what we know (we know, what we don't know we know, known unknowns etc... ) This one is going to be a fair deal tougher than just going out and collecting published work, citing some references and writing it up. We're in a position where we need to do some basic science before we can choose our targets (let alone hit them).

Chris' outline points are good ones, as are Hondero's up-thread. I think we'll get there, but it is going to take some time. If all the pieces that we're talking about come together this is going to be a heck of a seminal reference.

Matthias

OPTIMAL WEIGHT OF THE PROJECTILE

The common sense says to us that the heavier is a sling projectile at smaller speed we will be able to throw it and smaller range it will get, reason why we will use light projectiles to increase the range. It is the effect of "greater reach by speed".
But on the other hand, if we throw several projectiles of different weight at the same speed, the heaviest will reach greater distance since its stored energy will be greater, and the air drag will take more time in restraining it. It is the effect of "greater range by weight". As we see there are two opposed effects dealing with the weight, being more important one or another according to the weight that we are considering. If we suppose that we are using all our power of launching, an optimal weight will exist over wich the range will be smaller due to the superiority of the factor "greater range by speed", and also if the weight is below it, the range wiil be smaller too due to the factor "greater range by weight".
Evidently this optimal weight will be different for each slinger, since his muscular power will be different in general, and different therefore the relation weight-speed that he can obtain. This relation weight-speed of the projectile is related in the first place to the muscular behavior, it is to say to the own relation force-speed of contraction that the muscle is able to develop. This relation is expressed by the Hill´s formula, whose graphical representation is this:




As we see, as greater is the contraction speed of the muscle, less  is the force that it can develop. Evidently, both the speed and the force of contraction  will be transmited in certain proportion through the osseous system of the arm to the sling. One could think that the muscle is able to always develop the same maximum power, it is to say W=F*V = constant. Thus a diminution in the speed would make a proportional increase in the force. But the reality is more complex, like it can be observed drawing up the curve of power from the curve force-speed, that would be this:



We see that the maximum power is obtained for contraction speeds very reduced, of the order of 30% of the maximum speed. Nevertheless this fact does not worry us too much since in the sport we are considering we are located in the last third of the curve, that can be considered straight in first approach.

We have landed in the biomechanics. The arm, and in general the bones that take part in the launching are driven by muscles. The basic mechanical scheme is that of a lever leant on a joint and driven by a force applied near the joint that makes move the distant end and take to it a demultiplied force. It´s somehow like a little staff-sling. But in addition, the slinger model of propulsion is a set of conected levers that act in sequence, well coordinated, adding speed over speed to the hand. The hand, in turn, communicates this speed to the sling. And the question would be if we can substitute this compound lever for a single one at the time of making calculations to simplify them.

Well, this is the general exposition of the optimal weight problem. And for today it is enough  ???. It will continue.

PART 2

EFFICIENT WAY OF SLINGING


As Hondero mentined, Mother Nature has bestowed us with long bones of hand and tendons attached close joints. It has let us to make the rapid movements of hand and made us a quite good throwers, but, of not too heavy stuff. Fortunately, we have got also strong muscles of legs and trunk, of which we have learned use for more efficient throwing, over centuries of huntings, wars, plays and sports. So, there are the tested and efficient ways of throwing different things.

Therefore, it will be quite resonable, if our slinger would use one of these ways of throwing just for slingnig. For instance, the basebal pitcher's way (maybe, a bit modified, adapted to a sling motion).

Now, I try to explain how it working, using the simplified model.

Usually, the pitcher's palm is accelerated along some curved trajectory, because it mostly is a result of interfering rotatory motions of the hips, shoulders, arm, forearm and wrist. The acceleration surely changes along that composite curve.

For our purpose, let's assume, that that trajectory is circular and is a half of the full circle (180 degs).

Before we start our investigation of that throw, we should understand, how the cords behave during either a uniform rotation or eccelerated one, what forces act on a sling. That lets us to imagine how the trajectory of the stone looks.

Forces during a uniform rotation of the sling.

Here is the picture showing the forces during uniform rotations of the sling in the vacum (the air drag and gravity omitted):



Pic. 2


Practically, there is only a one external force, just the Centripetal Force, needed to keeping the circular trajectory of the stone (the Centrifugal Force isn't an external force, is only an inertial reaction to the curving the stone trajectory by the Centripetal Force).  

So, only job the palm must do, is pulling the cords with the Centripetal Force during the uniform whirling. No perpendicular to the radius, a driving force (which gives a torque) is needed, because there isn't an acceleration and air drag). After the release, the stone will start its stright travel along a tanget direction, with the same linear velocity, that had during the rotations (we know it from the Newton's First Law).

Angular acceleration = moment of interia * torque = 0  ---> torque = 0

In this case

radius of the stone orbit = radius of the palm orbit + lenght of the sling



Forces during an accelerated rotation of the sling

Now the case is a bit more complcated. When the palm accelerates, the cords become deviated to backward. The most short way of explanation, how it comes up to that deviation is, that a inertia of the stone (its mass) restrains its acceleration (First Newton's Law), saing simply, the stone "tries to delay itself". A deeper analysis of it would be very interesting, however for our purpose we skip it, as it is a bit complicated. Let's say only, the angle of the cords deviation results from a balance of forces. The picture below shows just these forces.



Pic. 3

Despite of common sense, tense sling works very similar to a stiff and light stick with a stone on the tip. But, there is one diference. We are able to provide the additional significant torque to a stick, by a suitable action of the fingers and wrist. In case of sling, it is almost impossible, bacause cords can't transfer any torque, due to its flexibility. In the other words, we are able to make the single stroke with stick using only our fingers and wrist, but try to do that with a relative long and sagging statically sling!
       
A stick is able to transfer either the torque or pulling or even pushing force, but cords only the pulling force! However, if one attaches a stick to fingers by a soft and short loop (or simply keeps the stick only beetwen the thumb and index finger), a stick would work almost exactly like a sling. In both cases, his palm only can pull these ones!

Consider, that just the pulling force is the only possible kind of interaction between slinger's palm and a stone! For lashing with a sitck, it is the strongest driving factor too, the figers and wrist action adds only some edge, in comparison to a sling.

So, there is the questions, how it is possible, that we are able to make so dynamic and spacious lash of a sling, only by pulling of the cords?

Look at the above picture (Pic. 2). During the acceleraton, the Pulling Force (Fp) transfered by the cords to the stone,  has the two component forces: the tangential one (Fa) and the radial one (Fcp). The first one (Fa) just transfers the torque to the stone and make it accelerated. The second one is the Centripetal Force, which keeps the stone on the curved trajectory. The forces Fai and Fcf are the interial reactions, the Fa "tries to restrain" the speeding up the stone, Fcf is the Centrifugal Force, which "tries" to make the stone trajectory just stright. They give the resultant force, the Fpi. Just this inertial resultant force (Fpi) determines the angle of the cords deviation from the radial direction, and a value and direction of the Pulling Force of the palm (Fp).

The mentioned forces are in a dynamic balance. The Fdi "tries" to deviate the cords to backward, but the Fcf "tries" to push them outside (erect them). They are changing their direction and value during the stone acceleration. It coerces a continous change of the cord deviations (the angle FI) and value of the Pull (Fp), acting on our fingers.

From the below formula (for the centrifugal force):

Fcf = [mass * (tangent velocity)^2]/radius

We see, that a value of the Fcf force is strongly dependend on the tangent velocity (^2). Therefore, that one must become higher and higher, since the stone is accelerated by the (Fa) force all the time. It causes, that the stone tends to go outside, what results in the additional rotation of the sling around the palm. As a result, the motion of the stone is compounded of the accelerated stone rotation around the centre of the circular orbit of the palm (our earlier assumptions), and of the accelerated rotation around the the palm (grip point).

The next picture shows what's going on.



Pic. 4

This is a quite likely sample, how the sling can run during a efficient stroke in a real conditions. W can see, that the stone accelerates more and more just in the last phase of the stroke, due to its high velocities (Fcf grows like tangent velocity^2) and due to decreasing the tangent stone acceleration (Fa decreases, bacause the arm and wrist become more and more erected, and additionaly more laden by the big Centrifugal Force). We can compare the Fcf to a (more and more stiff) spring, which is indirectly diffracted by the Fa, and which gives the kinetic energy back just before the release. That gives an additional spur for the stone, and makes a shape and lenght of the stone trajectory more beneficial, bacause lets us to release the stone in a much more profitable moment (I mean, more to forward, when the slinger's arm is alredy streched). You can see it on the above picture, that trajectory is a kid of a spiral. Of course, it isn't only possible variant in practice. It depends on a slinger's motion, his quickness, a leght of a sling, mass of a stone, and a few more factors. But, let's skip discussing it for the present, bacause, now we should start explore the adventage of our sling.


The adventage of the mentioned slinging way

Now let's play a bit with numbers again, and check what that adventage could be. The folowing analysis arises out of the assumption, that proceeding of the sling on the picture (Pic. 4) is similar to a real one during an efficient stroke.

Our assumptions:

R = 0.6 m              - radius of the palm circular trajectory;
                               (an result of the entire body motion);
L = 1 m                  - lenght of the sling;
THETA = 180 deg   - angle traced by the palm along the trajectory;
Vpi = 0                   - initial linear velocity of the palm;
Vsi = 0                   - initial linear velocity of the stone (assumption,
                               that the stone starts from the immobility is unrealistic,
                               especially for longer slings, because for doing an efficient stroke
                               we need some initial velocity for a proper tension of the cords);
t = 0.15 s               - time of the stroke (acceleration);

At my guess, that time practically could be even almost 50% shorter. This is the next parameter, we need to measure in a practice.

Vpf                 - final linear (tangent) velocity of the palm;
Vsf                 - final linear (tangent) velocity of the stone;
ap                  - mean tangent (along the trajectory) acceleration of the palm;
                      (constant equivalent of a variable acceleration);
as                  - mean tangent acceleration of the stone.

               
Let's calculate the mean tangent acceleration of the slinger's palm:

ap = 2*(S - Vpi*t)/t^2        

where Vpi = 0

and

Sp = 2*pi*R*(THETA/360)    - lenght of the palm arc

Sp = 2*3.14*0.6*(180/360) = 1.88 m

we get

ap = 2*1.88/0.15^2 = 167.1 m/s^2

Now we are able to calculate the final velocity of the palm:

From

ap = (Vpf - Vpi)/t         where Vpi = 0

we get

Vpf = ap * t = 167.1 * 0.15 = 25 m/s

Well, that velocity doesn't seem to be exaggerated, even for an average thrower.

Very interesting your analysis, Yurek!!. I think the position of the sling at the final stroke is quite accurate, though maybe it depends a little on the style you use. I´m addicted to accuracy and so my final stroke is very similar to a particular baseball pitch, describing the hand an almost straight trajectory from behind and ending with the arm streched towards the target. Thus the sling is more retarded at first and at the end it describe this aditional turn around the hand you say in a marked way. Other styles like Newmanitou or other guy whose name I don´t remember and that posted a good video, semms to have a snap with the sling almost like a continuation of the arm, in the upwards position.

The subjet is very interesting though it isn´t of general audience, but I  have to confess that I´m hooked.
I´have to modify a little my previous calculations considering this angle of the sling and the arm at the release in the general throwing style  .

OPTIMAL WEIGHT OF THE PROJECTILE (Final part)

We have seen that for the average slinger, throwing with spherical stone projectile, the optimal weight was of 55 gr. Nevertheless there are not much difference using others near weights. In fact, between 50 and 70 gr. the difference in range is smaller of 2%, barelly about four meters, which almost makes indifferent the use of anyone of these projectiles since other factors as small differences in the shape, rugosity of their surface or position of the projectile in the flight can compensate this difference.
I´m going to define a parameter that I´ll call 2% RWI (Interval of Weights for Range variations of 2 %). For the average slinger this parameter would be then 50-70.

Now let us see the behavior of the lead projectiles, also of spherical shape at first,  in the hands of the average slinger:

weight (gr).....velocity (m/s)......range (m)
   20...................... 66.66................... 255
   30...................... 65.27....................262.5
   40...................... 63.88....................264
   50...................... 62.5..................... 262
   60...................... 61.11................... 258



We see that now the optimal weight is 40 gr. and the 2 % RWI is 30-50. The light lead projectiles are then more advantageous, and in addition we even see that the famous 20 grams small glandes reach a distance smaller only a 3,5 % than the optimal weight, which explains its frequent presence among archaeological glandes.

We have spoken until now of the optimal weight for the average slinger... but will be it the same one for a more powerful slinger? The common sense says to us that existing enough throwing power, the heavy projectiles are more advantageous due to their greater dynamic inertia and greater difficulty to be restrained by the air. The results of the simulation for a slinger with speed in the empty hand of 130 Km/h and a static force in throwing position of 10 kg are:


weight (gr).......velocity (m/s)........range (m)
    50....................... 85.76....................391
    60....................... 84.86....................395.8
    70....................... 83.95................... 399.2
    90........................82.15................... 401
  100....................... 81.25................... 401.4
  120....................... 79.44................... 398.2
  130....................... 78.54................... 396
  150....................... 76.73....................389


We see that the optimal weight is now of 100 gr. and the 2% RWI are 60-130. The evident conclusion is that the more powerful is the slinger more advantages will have using heavy projectiles, although we see that the margin of weights is rather wide and that it will be more important than the optimal weight the other mentioned factors that take part in the range, as the shape, the rugosity of the surface and the position of the projectile in the flight. These points will be the following ones to analyze (HELP, Matthias...  )