Answer to Question #86791 in Mechanics | Relativity for Abel

Question #86791
The attacking ship has manoeuvred to 2500 m distance away from the mountain peak (1800 m high) and can shoot projectiles with an initial speed of 250 m/s. The shoreline on the right side of the mountain is horizontally 300 m away from its peak. Then the attacking ship has manoeuvred to take a new position to be at 2250 m away from the mountain peak. What are the distances right side of the mountain at which the defending ship can be safe from the bombardment of the attacking ship?
1
Expert's answer
2019-03-25T08:55:30-0400

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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