Question

3. You are an artillery officer attached to Gais Julis Caesar's Xth Legion in the Gallic War 1st 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 its projectile at speed U,-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 180. m, and the angle at which you must fire to achieve this.* Use g 9.81 m/s2 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 horizontal distance x = V_x t

x = V_o cos $\theta$ t

t = x / V_o cos $\theta$

and y = V_oy t - 1/2 g t²

y = V_o sin $\theta$ (x / V_o cos $\theta$ ) - g/2 ( x / V_o cos $\theta$ )²  

y = x tan $\theta$ - gx² / 2V_o² - g x² tan² $\theta$ /2V_o²

and dy/d $\theta$ = 0

d ( x tan $\theta$ - gx² / 2V_o² - g x² tan² $\theta$ /2V_o² ) / d $\theta$ = 0

x sec² $\theta$ - gx² tan $\theta$ sec² $\theta$ / V_o² = 0

( x - ( (gx² tan $\theta$ ) / V_o² )) sec² $\theta$ = 0

x - ( (gx² tan $\theta$ ) / V_o² = 0

tan $\theta$ = V_o² / gx

y = x tan $\theta$ - gx² / 2V_o² - g x² tan² $\theta$ /2V_o²  

y = V_o² / 2 g - gx² / 2 V_o²

y = ( 60² / 2 X 9.8 ) - ( 9.8 X 180 / 2 X 60² )

y = 183.673 - 0.245

y = 183.428 m