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W = F.dx = ma.dx = m.(dv/dt).dx = m.dv.(dx/dt)=m.dv.v = mv.dv. At t=0, x=0 and v=0. Now, F is applied for dt duration. At the end of dt, x change from 0 to dx and v changes from 0 to dv. You
are basically saying that v can be treated as dv all along. so, mv.dv becomes m(dv)(dv) or mv.v because both are same. Well, NO! v was 0 in the beginning and became dv only at the end of
dt. So, it is a triangle, not a rectangle. And, that's why you have 1/2, which matches with the integration of v.dv. So, 1/2 is correct.
