**ROUGH PROJECTILES **Although it´s somewhat incredible, the rugosity of the projectile has to do with the range, in the way of the golf balls, extending the reach, although not as much as in them because golf balls have less density and are affected in greater measure by this effect. Let us consider first a stone spherical projectile.

Re = Air density * Velocity * Diameter/air viscosity

Considering always the same projectile

**Re= K1 * V **On the other hand :

Drag force Fd = Cd/2 * air density * cross section * V^2 (CD = drag coefficient)

For the same projectile:

**Fd = K2 * CD * V^2 **The drag force will vary then with the speed of the projectile, first by itself and secondly by its dependency of CD that varies with Re, that varies as well with V. The relation of CD with Re comes given from experimental form by the curves of Achenbach for spheres:

The upper curve, corresponding to smooth spheres, has an almost constant value around 0,5 until arriving at the point in which the speed of the projectile causes a turbulent air flow, falling drastically the CD. Nevertheless, the required speed for this it is higer than the maximum one that can reach a slinger, reason why hardly we will enter this favorable zone.

Nevertheless, if the sphere is rough, we see that the corresponding curve provides lower values of CD, now in the margin of speeds of the slinger. For that reason the air drag force will be smaller and greater the range. The curve shown corresponds to a projectile of rugosity K/D = 0,01250, wich expresses the ratio between the depth of the holes and the diameter of the projectile. Thus, for a projectile diameter of 4 cm the holes would be of 0,5 mm. A projectile like this is easy to make striking a smooth-stone around all their surface with another stone, or with a rounded hammer.

In order to calculate the increase of range of this rough projectil vs a smooth or polished one we will have to make approximated calculations, since the speed of the projectile will be changing as it goes being restrained by the air, and therefore will be changing its CD. In the picture I have selected two zones that corresponds to the variation of speed for projectiles thrown with two initial speeds: 55,55 and 120 m/s, that are the speeds of launching of the considered average slinger and of a recordman.

I have used the Simulator to get in both cases an aproach to the end speed of the projectile, that could be of about 25 m/s and 65 m/s respectively. As it is very awkward to evaluate the effect of the changing Cd, I have taken an average Cd for each zone, as it is indicated in the figure. For a projectile of 100 gr. the calculated values of the range are:

Speed.......................55,55...........120 m/s

CD (Smooth) ........... 0,5............... 0,47

CD (rough)............... 0,272............0.37

Range (smooth )......167............... 353 ms

Range (rough )........ 208............... 409

We see that for the average slinger the

**increase of range with rough projectile is of 25%**. The recordman, nevertheless, is penalized by the high end speed of the projectile, that does not take advantage of the zone of lower Cd.

The following calcualtions will be with lead projectiles, that surely also are influenced by this rugosity effect (I progress little by little, but soon I´ll discover the carefully kept secret of Jurek... but I shall not disclose it

)