# Interweaving the Numerical Kinematic Symmetry Principles in School and Introductory University Physics Courses

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## Abstract

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## 1. Introduction

## 2. Accounting for Air Resistance

#### Atmospheric or Barometric—Choose Your Flavor

## 3. So Which Angle Is Best?

## 4. Conclusions

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Simulation of trajectories, factoring in drag force (left panel). The air density is constant in this case. On the right we see the maximal firing distance is given by firing in a more shallow angle (~33 degrees) than the canonical 45 degree angle.

**Figure 3.**Temperature and logarithmic atmospheric air density profile as function of height. We notice the differences between the naive atmospheric equation and the more detailed barometric equation diverge around 40 km above the air surface, for the initial surface temperature of 288.15 K.

**Figure 4.**While trajectories in vacuum (left panel, largest trajectories) reach a staggering firing distance of $>250\text{}\mathrm{km}$, introducing a constant air density greatly decreases the firing distance (left panel, smallest dot-dash). A realistic atmospheric profile yields trajectories that reach about 130 km (right panel). The difference between using an exponential density, and the more detailed result of the barometric equation is small, around tens to a few hundred meters.

**Figure 5.**Maximum firing distance as a function of the firing angle. This was simulated with a ground temperature of 288.15 K, and using the realistic atmospheric profile. Rather than a favoured ${45}^{\circ}$ angle for maximum distance, the optimal angle is ${52.7}^{\circ}$.

**Figure 6.**Optimal angle vs. Ground temperature suggests a linear dependence. This is due to $\rho \propto {T}^{-1}$ and the linear approximations we have made. The maximal height is also linear in temperature and is inversely correlated with the air density at the highest part of the trajectory.

**Figure 7.**Also shows the dependence of the firing range on the muzzle velocity. It should not surprise us, that there is a quadratic relation between the muzzle velocity and the firing range. However, whereas in the ballistic case with no drag, the relation is purely quadratic, i.e., of the form $\mathrm{R}={\mathsf{\alpha}\mathrm{v}}_{0}^{2}$, where $\mathsf{\alpha}$ is some coefficient, the drag term introduces a shift symmetry. The range equation is now given by:

**Figure 8.**Simulated data of optimal angle vs. projectile mass suggests that in the high mass limit the air density becomes negligible. Probing the 50–150 kg range alone would have resulted in a different apparent quadratic fit. However, that fit misses the high mass limit.

**Figure 9.**A wider view of the optimal angle as a function of mass, along with the percentage of energy loss during flight. This graph reveals that, indeed, the energy loss trends to zero where the large mass limit is taken.

**Figure 10.**A study of the impact velocity suggests the impact velocity goes to the initial velocity at the large mass limit. An exponential function is apparently a good fit, with a square-root of the mass as the argument. The right panels show the residual error (upper right), and the relative error (lower right). These show the numerical errors are of the order of ~2% or less where the function is well fitted.

**Figure 11.**The Paris Gun, and the Heavy Gustav. The Gustav, a much heavier super gun, firing a much heavier shell has reduced firing range, but due to being less susceptible to atmospheric (and other) effects is more accurate.

Ground Temperature (K) | Projectile Mass (kg) | Effective Static Drag Coefficient | $\mathbf{Elevation}\text{}(\circ )$ | Projectile Velocity (m/s) |
---|---|---|---|---|

288.15 | 106 | 0.21 | 45 | 1640 |

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## Share and Cite

**MDPI and ACS Style**

Ben-Abu, Y.; Yizhaq, H.; Eshach, H.; Wolfson, I.
Interweaving the Numerical Kinematic Symmetry Principles in School and Introductory University Physics Courses. *Symmetry* **2019**, *11*, 148.
https://doi.org/10.3390/sym11020148

**AMA Style**

Ben-Abu Y, Yizhaq H, Eshach H, Wolfson I.
Interweaving the Numerical Kinematic Symmetry Principles in School and Introductory University Physics Courses. *Symmetry*. 2019; 11(2):148.
https://doi.org/10.3390/sym11020148

**Chicago/Turabian Style**

Ben-Abu, Yuval, Hezi Yizhaq, Haim Eshach, and Ira Wolfson.
2019. "Interweaving the Numerical Kinematic Symmetry Principles in School and Introductory University Physics Courses" *Symmetry* 11, no. 2: 148.
https://doi.org/10.3390/sym11020148