Work done by a Gaussian impulse

This paper was first motivated by the following question: 'A pair of twins, R and S, each gives the same hard push on a block. R's block is on a rougher floor than S's. Who does more work?' It is shown that S will do more work on his block if there is no constraint on the distance over which the force is applied. On the other hand, if the assumption is relaxed to 'the same hard push within some distance', say due to the length of their arms, then it is possible that R does more work than S. To resolve this issue, general formulae for the velocity, displacement and work done are obtained for a Gaussian impulsive force acting on a body moving on a horizontal surface.