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It seems to me that it involves as much calculus as deriving
(1/2 mv^2).
This is usually done (in an algebra course) by stating that if v
increases linearly with time then v_avg = (v_i + v_f)/2.
If
this hand-
waving works, then we might as well say that if F increases linearly
with distance then F_avg = (F_i + F_f)/2, no?*
Is this hand-waving somehow worse than the hand-waving
performed to get
v_f^2 = v_i^2 + 2 a Delta x?
*[since F = kx, then
F_avg*Delta x = k (x_i + x_f)*(x_f - x_i)/2 = 1/2 k Delta (x^2).]