Let us use notation: "L = 2 m" - length of the barrel, "m = 23 kg" - mass of the object, "S = 1km = 1000m" - horizontal range of the projectile, "T = 5 s" - time to cover the distance "S".
Let us assume that the object is moving inside the barrel with constant acceleration, until at the end of the barrel it reaches the initial speed of the projectile "v_0", needed to cover the distance "S". The force needed to accelerate the object inside the barrel in that way is "F = \\frac{\\Delta p}{\\Delta t} = \\frac{m(v_2 - v_1)}{\\Delta T} = \\frac{m v_0}{\\Delta t}", since the object changes the momentum from to "mv_0".
Since the horizontal distance, covered in time "T" is "S = v_0 T", the initial speed is "v_0 = \\frac{S}{T}".
For accelerated motion inside the barrel, the following equations hold: "L = \\frac{a (\\Delta t)^2}{2}", "v_0 = a \\Delta t". Dividing the first equation by second, obtain "\\frac{L}{v_0} = \\frac{\\Delta t}{2}", from where "\\Delta t = \\frac{2 L}{v_0}" - the time of the motion inside the barrel.
Hence, using formula for "\\Delta t" and "v_0" , obtain "F = \\frac{m v_0}{\\Delta t} = m \\frac{v_0^2}{2 L} = \\frac{m}{2 L} \\frac{S^2}{T^2} = 250kN".
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