Veneficus Angli: The Magic Angle

Grace KohnMickey Mulder

Nick SatherAmaya Bayne

Research Question

• What is the angle at which, when launched from a slingshot, with all other variables kept constant, an object will fly the greatest distance?

History of the Slingshot• It is unknown where the slingshot first

originated, some say the first slingshots came from Russia, others say that the slingshot came from as far back as the prehistoric times. However one thing’s for sure, the slingshot came from the ‘sling’, a projectile weapon typically used to throw stones. It is also known as the Shepherd’s sling. Ancient peoples used the sling in combat and organized their armies with a section of specialist slingers. The slingers were usually put in the back as backup. The sling had many great advantages as a weapon, it was light weight and easy to carry and extremely effective. A sling bullet lobbed in high trajectory can achieve ranges approaching 600m, which was impressively farther than what could be achieved by bows in any period. The sling was also used by shepherds to fend off other animals, as well as for hunting small animals such as rabbits and birds.

History of the Slingshot (continued)

• The sling is mentioned in the Bible, believed to be the oldest textual reference to a sling. It is certain that slings were known to Neolithic people around the Mediterranean and it is possible that sling could have been invented during the Upper Paleolithic times, the third and last subdivision of the Old Stone Age, A time when new inventions such as the bow and arrow were emerging. However, once again, nothing is certain. The sling became common all over the world, but it is also unclear whether this occurred because of independent invention or perhaps cultural diffusion.

Background Information• We made our hypothesis based on background research which can

be summed up in this excerpt:– Launching a projectile at 45 degrees from the ground will allow it to

reach it's optimum distance (in the x axis).

This is because the initial velocity is evenly distributed between the x and y directions. Giving the projectile the most air time with the highest horizontal velocity.

– The above is true for a projectile launched in a vacuum so that there are no drag forces. The drag forces depend upon both mass and shape."Because the horizontal component of a projectile's velocity decreases steadily as it moves through the air, its path is not symmetric about its highest point. .... in real life --the maximum range occurs for an initial angle of less than 45o. The greater the drag force, the lower the angle for Rmax (maximum range). For a baseball struck hard by a bat, the optimum angle is about 40o." [Beiser, A. Physics, 5th edition, p52]

Hypothesis

• We believe that the angle at which the greatest distance will be reached will be 45°.

Materials

• Slingshot

• 2 Giant Protractors

• Tape Measurer

• 2 Wooden Balls

Set-Up

• The Slingshot: – We took two 45° protractor ramps and attached them together

with screws, so we could emulate a slingshot that could reach above our hypothesized angle. Next we drilled two screws into the top of the upper protractor ramp, so as to act as holders for each end of the rubber band sling.

• The Sling:– The sling consisted of a giant rubber band, with a leather strip

attached that acted as cradle for the wooden ball.

• The Ball: – The ball, a 2 gram wooden sphere, was painted in an even coat

of neon pink paint in order to help see where it landed. The paint added a minimal amount of mass to the ball.

Set Up

Procedure• Place launcher on flat level surface• Place rubber band on launcher• Weigh ball• Launch ball at 15°,30°,45°, and 60°

– Place ball in launcher– Pull back rubber band and leather strip to the back of the launcher– FIRE!!!

• Record distances• Launch ball at distances in between to help pinpoint exact distances (For

example 30°, 35°,40°, 50°, 55°)– Place ball in launcher– Pull back rubber band and leather strip to the back of the launcher– FIRE!!!

• Record distances• Launch ball between the angles shooting the furthest distances, so as to

find the exact angle. (For example if the furthest shooting angles are 35° and 40°, launch at angles 36°, 37°, 38°, and 39°)

– Place ball in launcher– Pull back rubber band and leather strip to the back of the launcher– FIRE!!!

• Repeat steps 1-8 for trails 2 and 3.

Photos of Experimentation

DataA° =AngleD=Distance (rounded to the nearest inch)

Angle Distance (T1) Distance (T2) Distance (T3) Average Distance

15 436 449 424 436.33

30 585 748 793 708.66

35 758 762 766 762

40 1527 1372 1059 1319.33

41 1415 1362 1378 1385

42 1486 1401 1348 1411.66

43 1285 1337 1423 1348.33

44 1147 1359 1221 1242.33

45 935 1380 1212 1175.66

50 892 821 855 856

55 682 834 698 738

60 417 505 483 468.33

Data Analysis• Our data table showed that an angle of 42° shot the greatest

average distance, with a max distance of 1,487 inches or about 124ft, and a minimum distance of 1,348 inches or about 112ft. However, the furthest single shot occurred at an angle of 40° with a distance of 1,524 inches, unfortunately that result couldn’t be duplicated for a second or third time. The shortest single shot was at an angle of 60° with a distance of 416 inches, less than a third of the distances recorded at 42°. We observed that as we shot the ball at a higher angle the height of the shot seemed to increase, resulting in a more direct spot of impact with no roll, were as at low angles the ball would still land at a short distance but would roll for a long ways. We also noticed that at the higher altitudes such as in 55° and 60° the ball would severely slow down and than ‘float’ back to the ground. This may be a result of our ball only having a mass of 2 grams.

Conclusion• After completing numerous trials of our experiment and analyzing

the data, we came to a conclusion that answered our research question: what is the angle that, when launched, with all other variables kept constant, will an object travel the furthest distance? Our hypothesis, which was that the theoretical angle of 45° would create the furthest distance of the launched object, in this case a ball, was not correct. We determined that the angle that would allow for the furthest distance was somewhere between 40° and 45°. The magic angle seemed to be 42°. We did not have a way to measure wind gusts or resistance because of the primitive nature or our contraption. Because our results do not take into account measurements of air resistance, they are not as precise as they could be. An interesting way to interpret the data could have been the correlation between the air resistance and the distance traveled. However, though we do not have these measurements we can infer the effects the air resistance had on our results.

Conclusion (continued)• The air resistance played a role in making

distance the ball flew when launched from 45° less than the distance when the ball was launched from a lower angle. Thus, we concluded that when air resistance for our experimental conditions is taken into account, the angle that provides for the farthest distance is about 42°. Because of the theoretical angle and the results that we found, we conclude that depending on air resistance, the angle at which an object would fly the furthest would between 40° and 45° depending of the air resistance of the conditions in which the experiment is conducted.

Conclusion (continued)

• The angle that we found makes sense in comparison to the theoretical angle because if the ball was released at a lower angle it would have less air resistance acting on it (because it is in the air for less time) than if it was shot at higher angle.

• Overall, the application of this experiment to the larger topic of warfare is applicable; however, there is not one answer for the magic angle. If a weapon is to be launched it is best between 40° and 45°. The weapon will have best results if the air resistance is measured and taken into account. However, since this is not a practical procedure, the success of the weapons’ launching distance may not be maximized to its fullest extent, but if the range of these angles is used, the distance can be extended.