A particle moves along a straight line with uniform acceleration $a$. It covers distances $s_1$ and $s_2$ in consecutive intervals of time $t_1$ and $t_2$. Prove that the acceleration is given by $a = \frac{2(s_2 t_1 - s_1 t_2)}{t_1 t_2 (t_1 + t_2)}$.

A body slides down a rough plane inclined at $30^\circ$ to the horizontal. If the coefficient of kinetic friction is $\mu_k = \frac{1}{2\sqrt{3}}$, find the acceleration of the body down the incline and the distance traveled in $4\text{ seconds}$ from rest ($g = 9.8\text{ m/s}^2$).

A train starts from rest at station $A$ with uniform acceleration $a = 0.5\text{ m/s}^2$ for $40\text{ seconds}$, then runs at constant speed for $2\text{ minutes}$, and finally retards uniformly to rest at station $B$ in $20\text{ seconds}$. Calculate: (a) The maximum speed reached by the train. (b) The total distance between stations $A$ and $B$. (c) The average speed of the train for the whole journey.