A ball is thrown vertically upwards with an initial speed $u$ from a height $h$ above the ground. The ball eventually hits the ground with a speed $v$. The acceleration due to gravity is $g$ and air resistance is negligible. What is the average speed of the ball over its entire trajectory?

The velocity $v(t)$ of a particle moving in one dimension is given by:

$$ v(t)= \begin{cases}\alpha t, & 0 \leq t \leq T / 3 \\ \alpha T / 3, & T / 3 \leq t \leq 2 T / 3 \\ \alpha(T-t), & 2 T / 3 \leq t \leq T\end{cases} $$

where $\alpha(\neq 0)$ is a constant. What is the displacement of the particle from time $t=0$ to $T$ ?

A particle experiences an acceleration $\vec{a}=\alpha \vec{v}$, where $\vec{v}$ is the velocity of the particle and $\alpha$ is a constant. If the distances traveled by the particle in the time intervals $t_2-t_1$ and $t_3-t_1$ are $S_{12}$ and $S_{13}$, respectively, which of the following relations is true?