Question

You are an artillery officer attached to Gaius Julius Caesar's Xth Legion in the Gallic War (1rst C. BCE). You are responsible for a mangonel, a device with a long spoonlike arm used to lob stones over the walls of Gallic oppida. These fortresses are protected by high walls atop a hill rising from a deep, wide moat: you must fire your missiles from the far side of the moat. Your mangonel can fire it's projectile at speed (Vo)= 60.0 m/s, independent of the angle of elevation at which you fire. Calculate the height of the highest wall you can clear from across a moat of width x=180.m, and the angle at which you must fire to achieve this. Use g= 9.81 m/s^2 as the acceleration of gravity. Assume aerodynamic forces are negligible and disregard the height of the mangonel itself. HINT: When aimed for maximum height, the projectile will not clear the wall at the top of its trajectory. A more careful analysis is needed.

Answer 1

The equations for the projectile are

$y\left ( t \right )= y_{0}+V_{0y}t-\frac{1}{2}gt^{2}$ __________________(1)

$x\left ( t \right )= V_{0x}t$ _______________________________(2)

where,

$V_{0x}= V_{0}\, cos\, \theta$

$V_{0y}= V_{0}\, sin\, \theta$

Using equations (1)-(2) we get,

$y\left ( x \right )= y_{0}+tan\, \theta \, x-\frac{g}{2V_{0}^{2} cos^{2}\, \theta }x^{2}$

the maximum x occur for $\theta = \frac{\pi }{4}$ , then let us try to find out the height, for that angle,

as

$y\left ( x \right )= tan\, \theta \, x-\frac{g}{2V_{0}^{2} cos^{2}\, \theta }x^{2}$

the maximum distance x can be achieve when theta=pi/4

$y\left ( x= 180\, m \right )= tan\, \left ( \frac{\pi }{4} \right ) \, \times 180\, m-\frac{9.81\, \frac{m}{s^{2}}}{2\left ( 60\frac{m}{s} \right )^{2} cos^{2}\, \frac{\pi }{4} }\left ( 180\, m \right )^{2}=91.71\, m$

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