1.46: Velocity-time graph and exponential velocity

The diagram shows part of the velocity–time graph for a particle, moving at $v$ m s$^{-1}$ in a straight line, $t$ s after passing through a fixed point. The particle travels at $U$ m s$^{-1}$ for $20$ s and then decelerates uniformly for $10$ s to a velocity of $\dfrac{U}{2}$ m s$^{-1}$. In this $30$ s interval, the particle travels $165$ m. Find the value of $U$.

Find the acceleration of the particle between $t = 20$ and $t = 30$.

A particle $P$ travels in a straight line such that, $t$ s after passing through a fixed point $O$, its velocity, $v$ m s$^{-1}$, is given by $v = \left(\mathrm{e}^{\frac{t^{2}}{8}} - 4\right)^{3}$. Find the speed of $P$ at $O$.

Find the value of $t$ for which $P$ is instantaneously at rest.

Find the acceleration of $P$ when $t = 1$.