3.1 Position, displacement, and average velocity  (Page 2/10)

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We use the uppercase Greek letter delta (Δ) to mean “change in” whatever quantity follows it; thus, $\text{Δ}x$ means change in position (final position less initial position). We always solve for displacement by subtracting initial position ${x}_{0}$ from final position ${x}_{\text{f}}$ . Note that the SI unit for displacement is the meter, but sometimes we use kilometers or other units of length. Keep in mind that when units other than meters are used in a problem, you may need to convert them to meters to complete the calculation (see Appendix B ).

Objects in motion can also have a series of displacements. In the previous example of the pacing professor, the individual displacements are 2 m and $-4$ m, giving a total displacement of −2 m. We define total displacement     $\text{Δ}{x}_{\text{Total}}$ , as the sum of the individual displacements , and express this mathematically with the equation

$\text{Δ}{x}_{\text{Total}}=\sum \text{Δ}{x}_{\text{i}},$

where $\text{Δ}{x}_{i}$ are the individual displacements. In the earlier example,

$\text{Δ}{x}_{1}={x}_{1}-{x}_{0}=2-0=2\phantom{\rule{0.2em}{0ex}}\text{m.}$

Similarly,

$\text{Δ}{x}_{2}={x}_{2}-{x}_{1}=-2-\left(2\right)=-4\phantom{\rule{0.2em}{0ex}}\text{m.}$

Thus,

$\text{Δ}{x}_{\text{Total}}=\text{Δ}{x}_{1}+\text{Δ}{x}_{2}=2-4=-2\phantom{\rule{0.2em}{0ex}}\text{m}\text{​.}$

The total displacement is 2 − 4 = −2 m to the left, or in the negative direction. It is also useful to calculate the magnitude of the displacement, or its size. The magnitude of the displacement is always positive. This is the absolute value of the displacement, because displacement is a vector and cannot have a negative value of magnitude. In our example, the magnitude of the total displacement is 2 m, whereas the magnitudes of the individual displacements are 2 m and 4 m.

The magnitude of the total displacement should not be confused with the distance traveled. Distance traveled ${x}_{\text{Total}}$ , is the total length of the path traveled between two positions. In the previous problem, the distance traveled    is the sum of the magnitudes of the individual displacements:

${x}_{\text{Total}}=|\text{Δ}{x}_{1}|+|\text{Δ}{x}_{2}|=2+4=6\phantom{\rule{0.2em}{0ex}}\text{m}\text{.}$

Average velocity

To calculate the other physical quantities in kinematics we must introduce the time variable. The time variable allows us not only to state where the object is (its position) during its motion, but also how fast it is moving. How fast an object is moving is given by the rate at which the position changes with time.

For each position ${x}_{\text{i}}$ , we assign a particular time ${t}_{\text{i}}$ . If the details of the motion at each instant are not important, the rate is usually expressed as the average velocity     $\stackrel{\text{–}}{v}$ . This vector quantity is simply the total displacement between two points divided by the time taken to travel between them. The time taken to travel between two points is called the elapsed time     $\text{Δ}t$ .

Average velocity

If ${x}_{1}$ and ${x}_{2}$ are the positions of an object at times ${t}_{1}$ and ${t}_{2}$ , respectively, then

$\begin{array}{}\\ \\ \text{Average velocity}=\stackrel{\text{–}}{v}=\frac{\text{Displacement between two points}}{\text{Elapsed time between two points}}\\ \stackrel{\text{–}}{v}=\frac{\text{Δ}x}{\text{Δ}t}=\frac{{x}_{2}-{x}_{1}}{{t}_{2}-{t}_{1}}.\end{array}$

It is important to note that the average velocity is a vector and can be negative, depending on positions ${x}_{1}$ and ${x}_{2}$ .

Delivering flyers

Jill sets out from her home to deliver flyers for her yard sale, traveling due east along her street lined with houses. At $0.5$ km and 9 minutes later she runs out of flyers and has to retrace her steps back to her house to get more. This takes an additional 9 minutes. After picking up more flyers, she sets out again on the same path, continuing where she left off, and ends up 1.0 km from her house. This third leg of her trip takes $15$ minutes. At this point she turns back toward her house, heading west. After $1.75$ km and $25$ minutes she stops to rest.

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