1.43: Simple harmonic motion and velocity-time graph

A particle $P$ moves in a straight line such that its displacement, $x$ m, from a fixed point $O$ at time $t$ s is given by $x = 10\sin 2t - 5$. Find the speed of $P$ when $t = \pi$.

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

Find the acceleration of $P$ when it is first at rest.

The diagram shows the velocity–time graph for a particle $Q$ travelling in a straight line with velocity $v$ m s$^{-1}$ at time $t$ s. The particle accelerates at $3.5$ m s$^{-2}$ for the first $10$ s of its motion and then travels at constant velocity, $V$ m s$^{-1}$, for $10$ s. The particle then decelerates at a constant rate and comes to rest. The distance travelled during the interval $20 \le t \le 25$ is $112.5$ m. Find the value of $V$.

Find the velocity of $Q$ when $t = 25$.

Find the value of $t$ when $Q$ comes to rest.