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[A] Calculate the value of the action integral between the limits $t=0$
      and $t=T$ for a a particle falling under influence of gravity along the
      following three paths.

 (i)  for a fictitious motion with path given by $z= at.$
 (ii) for a second fictitious path given by $z=bt^3.$
 (iii) for the real motion $z={1\over 2} g t^2. $

      where the constants $a, b$ must be determined so that the initial and
      final positions coincide with the rules of variation in the action

   [B] Check if  the action integral has smaller value for the real motion
      in(c) than the fictitious ones (a) and (b). Discuss the result you have
      obtained and write conclusions you may draw about the action principle.

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