1.30: Kinematics: velocity-time graphs and exponential motion

The diagram shows the velocity-time graph of a particle $P$ moving in a straight line with velocity $v\text{ ms}^{-1}$ at time $t\text{ s}$ after leaving a fixed point. Find the distance travelled by the particle $P$.

Write down the deceleration of the particle when $t = 30$.

The diagram shows a velocity-time graph of a particle $Q$ moving in a straight line with velocity $v\text{ ms}^{-1}$, at time $t\text{ s}$ after leaving a fixed point. The displacement of $Q$ at time $t\text{ s}$ is $s\text{ m}$. On the axes below, draw the corresponding displacement-time graph for $Q$.

The velocity, $v\text{ ms}^{-1}$, of a particle $R$ moving in a straight line, $t\text{ s}$ after passing through a fixed point $O$, is given by $v = 4\mathrm{e}^{2t} + 6$. Explain why the particle is never at rest.

Find the exact value of $t$ for which the acceleration of $R$ is $12\text{ ms}^{-2}$.

Showing all your working, find the distance travelled by $R$ in the interval between $t = 0.4$ and $t = 0.5$.