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Recently, a friend and I were watching a 2nd world war film where a battleship was shelling land based fortifications. The discussion arose on how the ship (which was moving up and down due to the waves in the sea) could accurately target the fortifications.

I remember being told that the gunners waited until the ship was at the same state in the cycle before firing. This lead to duscussing what would be the best part of the cycle to fire. My view was that either the maximum or minimum of the cycle would be best, my friend thought the middle of the cycle would be better, as this give twice the number of opportunities to fire. I feel the only way to resolve this would be to calculate the positional probability distribution function for the gun on the ship undergoing simple harmonic motion.

I would appreciate help with this. Namely

Assuming the battleship is moving up and down on the waves where its vertical displacement about a mean is given by:

d = A sin (wt)

At any given arbitrary time T what would be the probability of finding the ship at a given displacement from the mean D.

Thanks

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# Simple Harmonic Motion and Battleships

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