Question

(a)A particle starts by moving to the right along a horizontal line; the graph of its position function is shown in the figure. When is the particle moving to the right? Moving to the left? Standing still?

(b) Draw a graph of the velocity function.

Short answer

(a)The particle is moving towards to the right direction when the speed \(s\) is increasing on the intervals \((0,1)\)and \((4,6)\).

The particle is moving towards to the left direction when speed \(s\) is decreasing on the interval \((2,3)\).

The particle is standing still when the speed \(s\) is constant on the intervals \((1,2)\)and \((3,4)\).

(b)

Step-by-step solution

(a) Step 1: Find the average velocity over time interval

From the graph it can be observed that the particle is moving towards to the right direction when the speed\(s\) is increasing on the intervals \((0,1)\)and \((4,6)\).

The particle is moving towards to the left direction when speed \(s\) is decreasing on the interval \((2,3)\).

The particle is standing still when the speed\(s\) is constant on the intervals \((1,2)\)and \((3,4)\).

(b) Step 2: Draw a graph of the velocity function.

As at the corner points on the graph there is no slope. Find slope on the interval\((0,1)\).

\(\begin{aligned}{\rm{Slope}} &= \frac{{3 - 0}}{{1 - 0}}\\ &= 3\end{aligned}\)

Find slope on the interval\((2,3)\).

\(\begin{aligned}{\rm{Slope}} &= \frac{{1 - 3}}{{3 - 2}}\\ &= - 2\end{aligned}\)

Find slope on the interval\((4,6)\).

\(\begin{aligned}{\rm{Slope}} &= \frac{{3 - 3}}{{6 - 4}}\\ &= 1\end{aligned}\)

Construct graph as: