3.118: Displacement–time graph and particle kinematics

In this question, all lengths are in metres and time $t$ is in seconds.

The diagram shows the displacement–time graph for a runner for $0\leqslant t\leqslant 40$.

Find the distance the runner has travelled when $t=40$.

On the axes, draw the corresponding velocity–time graph for the runner for $0\leqslant t\leqslant 40$.

A particle $P$ moves in a straight line such that its acceleration is $\left(t^{2}+4\right)^{-1/2}$ for $t\geqslant 0$. Given that $P$ has velocity $9$ when $t=6$, find the velocity of $P$ at time $t$.

Given that $s=\tfrac{1}{3}$ when $t=6$, find the displacement of $P$ at time $t$.