Problem
A particle $P$ moves on the positive $x$-axis. The velocity of $P$ at time $t$ seconds is $\left(2 t^{2}-9 t+4\right) \mathrm{ms}^{-1}$. When $t=0, P$ is $15 \mathrm{~m}$ from the origin $O$. It can be shown that $P$ is instantaneously at rest when - $\frac{1}{2}$ and when,$-\boldsymbol{}$. Find the total distance travelled by $P$ in the interval $0 \leq t \leq 5$ (5 marks)
Solution
Step 1 ：Integrate the velocity function to find the position function: \(x(t) = \frac{2}{3}t^3 - \frac{9}{2}t^2 + 4t + 15\)
Step 2 ：Find the position at t=0, t=0.5, and t=5: \(x(0) = 15\), \(x(0.5) \approx 15.96\), \(x(5) = \frac{35}{6}\)
Step 3 ：Calculate the total distance traveled: \(\boxed{11.08}\) meters
