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Distance And Height Of Cannon Projectiles

2891 words - 12 pages

Distance and height of cannon projectiles
IB Mathematics SL
Mathematical investigation

For the mathematical investigation, I want to focus on angles and trigonometry and differentiation, since these areas are the ones interesting me the most in the SL syllabus, and connect it to war history, and weapons used in wars, which are some of my other interests. I take history on higher level in the IB, and war has always been one of my interests.
The cannon has improved over the past 1000 years and has been an effective way for an army to shoot a projectile from a long range into an enemy fortress or behind enemy walls. But did the angle of which the cannons were fired at have any effect on how far the projectile would move, and were the soldiers steering the cannons aware about the angle that would move the projectile the furthest and be the most accurate.
Under the wars that occurred in the 19th and 20th century, there is a high possibility that the soldiers arming and shooting the cannon did not go through the appropriate training in how to optimize their accuracy. An example for this is the American civil war, where the soldiers could have used the “trial and error” methodology i.e. shoot once from a certain angle, see where it lands, and modify the angle according to distance from target. I started thinking about how the awareness of the correlation between speed and shooting angle could affect the efficiency of the army, since they would have to shoot less and by that safe time and money. By that investigate how different cannons with different velocities, could be an advantage for the soldiers. Furthermore, using calculus, I will investigate the maximum distance of various cannons used in the war. The question I want to explore is therefore if applying trigonometry and differentiation in figuring out the angle of shot will make the army more efficient.

In theory the cannons could be fired at an angle between 0 and 90° since anything else would cause in the projectile not moving forward and towards the target. In practice, the range would be much more narrow.
By using trigonometry, you can calculate the distance of the projectile, as long as you know the initial velocity and the angle of which the cannon is fired at. (see figure 1 below)

Figure 1: showing the angle of the shooting, the maximum height and the distance travelled, using the horizontal and vertical angles

Below (table 1) are a variety of cannons used in the wars. Using the velocity plus a predefined angle, I will be able to calculate the distance the projectile will travel. These cannons were the most commonly used, some were developed in the civil war, while others had been around for some years but were still in use.

Name of Cannon Weight of Projectile
(kg) Velocity
(m/s)

6-pounder Gun 2.77 438.61
M1857 12-pounder "Napoleon" 5.58 438.9
12-pounder Howitzer 4.04 321.26
24-pounder Howitzer 11.56 323.09
10-pounder Parrott rifle 2.67 374.9
3-inch Ordnance Rifle 2.67...

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