Chapter 11.2, Problem 9QR

To determine

To find: The sketch for the curve over the given $t$ interval indicating the direction in which it is traced.

Answer to Problem 9QR

The total distance travelled by the particle from $t=0$ to $t=4$ is 32 units to the left of the x axis.

Explanation of Solution

Given Data:

The particle moves along the x axis with the velocity $v\left(t\right)=3t^2-12t$ for $t\ge 0$ then the total distance travelled by the particle from $t=0$ and $t=4$ .

Calculation:

Consider the given equation is,

  $v\left(t\right)=3t^2-12t$

Then, the displacement of the particle from $t=0$ to $t=4$ is,

  $\begin{array}{c}x={\displaystyle {\int }_0^4\left(3t^2-12t\right)dt}\\ ={\left(3\cdot \frac{t^3}{3}-\frac{12t^2}{2}\right)}_0^5\\ ={\left(t^3-6t^2\right)}_0^4=\\ =-32\end{array}$

Thus, the total distance travelled by the particle from $t=0$ to $t=4$ is 32 units to the left of the x axis.