The angle of elevation of the shell must be approximately 25.840° in order to have a reach of 20 kilometers.
The shell experiments a parabolic motion, which is a combination of horizontal uniform motion and vertical uniform accelerated motion. If initial height of the shell is the same that its final height, then the range of the shell ([tex]d[/tex]), in meters, is described by the following expression:
[tex]d = \frac{v^{2}}{g}\cdot \sin 2\theta[/tex] (1)
Where:
If we know that [tex]v = 500\,\frac{m}{s}[/tex], [tex]g = 9.807\,\frac{m}{s^{2}}[/tex] and [tex]d = 20000\,m[/tex], then the angle of elevation is:
[tex]\sin 2\theta = \frac{d\cdot g}{v^{2}}[/tex]
[tex]\theta = \frac{1}{2}\cdot \sin^{-1} \frac{d\cdot g}{v^{2}}[/tex]
[tex]\theta = \frac{1}{2}\cdot \sin^{-1} \frac{(20000\,m)\cdot \left(9.807\,\frac{m}{s^{2}} \right)}{\left(500\,\frac{m}{s} \right)^{2}}[/tex]
[tex]\theta \approx 25.840^{\circ}[/tex]
The angle of elevation of the shell must be approximately 25.840° in order to have a reach of 20 kilometers.
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