A particle $P$ moves on the positive $x$ -axis. The velocity of $P$ at time $t$ seconds is $(2 t^{2} - 9 t + 4) {ms}^{- 1}$ . When $t = 0, P$ is $15 m$ from the origin $O$ .
It can be shown that $P$ is instantaneously at rest when - $\frac{1}{2}$ and when, $-$ .
Find the total distance travelled by $P$ in the interval $0 \leq t \leq 5$ (5 marks)

Answer

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Answer

Calculate the total distance traveled: $11.08$ meters

Steps

Step 1 ：Integrate the velocity function to find the position function: $x (t) = \frac{2}{3} t^{3} - \frac{9}{2} t^{2} + 4 t + 15$

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: $11.08$ meters