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See the image above. I am able to get to (where x = distance):

x = (1/2k)ln((g - k(V^2)(tan a )^2)/(g - kv^2))

If you let v = 0, and then multiply by mg, you find potential energy gained presumably - I dont understand why the question states the potential energy lost.

Obviously the 0.5(MV^2)*(tan a)^2 part of the expression is due to the lost of the kinetic energy, however for the other part of the expression I am struggling.

Thanks for any help in advance.

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# Motion of a particle subject to a variable resisitive force

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