Friction is a nonconsertive force and must be accounted for. Typically, air resistance is not a linear function of velocity. So, different amounts of energy are lost to air friction for the different paths of travel. To prove this, set up an equation of motion for each direction incorporating the frictional force and assume the applied force is a funtion of velocity.
x'' - bv2 + x = F(v)
where x'' is the second time derivative of x.
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