I don't know what you mean by "applying acceleration to produce a force". The only force that I can think of that can arguably do that is gravity, and to describe that you need general relativity. None of the other forces apply acceleration to an object independent of its mass. General relativity doesn't allow things to move faster than light either.
Also, I again encourage you to calculate what the momentum, and therefore the force, would be if v were greater than c. You're going to be taking the square root of a negative number...
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