By understanding the problem, the attacking ship is ultimately at the 2250 m distance from the mountain peak (horizontally).
We will denote the given values as:
"x_a=2250~\\text{m};~h=1800~\\text{m};~x_{shore}=300~\\text{m};~v=250\\frac{\\text{m}}{\\text{s}}. \\\\"
The trajectory of a projectile motion:
"y(x) = x\\tan\\alpha - \\frac{gx^2}{2v^2\\cos^2\\alpha}. \\\\"
By making sure that by the moment when a projectile reaches the mountain, it can pass over its peak, we can find the acceptable range of angles:
"y(x_a) = x_a\\tan\\alpha - \\frac{gx_a^2}{2v^2\\cos^2\\alpha} > h; \\\\"
We shall enter numerical values and resort to some calculation aid here:
In: Solve[2250*Tan[a] - (9.81*(2250)^2)/(2*((250)^2)*(Cos[a])^2) == 1800, a]
Out: {{a -> -2.2438}, {a -> -1.79385}, {a -> 0.897791}, {a -> 1.34775}}
By discarding the negative results and converting the results to degrees (for better illustration):
In: 0.8977911361271153*180/Pi
Out: 51.4396
In: 1.347746132891334*180/Pi
Out: 77.2202
we can see that projectile will pass over the peak if shot between the angles of 51.44 and 77.22 degrees.
Simple check of the maximum range "L = \\frac{v^2}{g}\\sin{(2\\alpha)}" will prove that the angle 51.44 degrees will give farther range:
In: (250^2)*Sin[2*0.8977911361271153]/9.81
Out: 6210.77
In: (250^2)*Sin[2*1.347746132891334]/9.81
Out: 2748.79
Therefore let's label "\\alpha_m \\approx 51.44\\degree. \\\\"
Maximum range, as already calculated, "L_{max} = \\frac{v^2}{g}\\sin{(2\\alpha_m)} \\approx 6211~\\text{m}."
So the safe distance for the defending ship, if calculating from the peak base, horizontally is
"x_{safe} = L_{max} - x_a \\approx (6211-2250)~\\text{m} \\approx 3961~\\text{m}. \\\\"
One can also obviously see that the projectile in this case will pass the horizontal distance of the shore (300 m).
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