1.48: Displacement and velocity graphs, exponential motion

The diagram shows the displacement–time graph of a particle $P$ which moves in a straight line such that, $t$ s after leaving a fixed point $O$, its displacement from $O$ is $x$ m. On the axes below, draw the velocity–time graph of $P$.

A particle $Q$ moves in a straight line such that its velocity, $v$ m s$^{-1}$, $t$ s after passing through a fixed point $O$, is given by $v = 3\mathrm{e}^{-5t} + \dfrac{3t}{2}$, for $t \ge 0$. Find the velocity of $Q$ when $t = 0$.

Find the value of $t$ when the acceleration of $Q$ is zero.

Find the distance of $Q$ from $O$ when $t = 0.5$.