[QUE/CM-02011]

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.

1. for a fictitious motion with path given by $z= at.$
2. for a second fictitious path given by $z=bt^3.$
3. for the real motion $z=\frac{1}{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 principle. Check if that the integral has smaller value for the real motion in(c) than the fictitious ones (a) and (b)