# 3.1 Position, displacement, and average velocity  (Page 3/10)

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1. What is Jill’s total displacement to the point where she stops to rest?
2. What is the magnitude of the final displacement?
3. What is the average velocity during her entire trip?
4. What is the total distance traveled?
5. Make a graph of position versus time.

A sketch of Jill’s movements is shown in [link] .

## Strategy

The problem contains data on the various legs of Jill’s trip, so it would be useful to make a table of the physical quantities. We are given position and time in the wording of the problem so we can calculate the displacements and the elapsed time. We take east to be the positive direction. From this information we can find the total displacement and average velocity. Jill’s home is the starting point ${x}_{0}$ . The following table gives Jill’s time and position in the first two columns, and the displacements are calculated in the third column.

Time t i (min) Position ${x}_{i}$ (km) Displacement $\text{Δ}{x}_{\text{i}}$ (km)
${t}_{0}=0$ ${x}_{0}=0$ $\text{Δ}{x}_{0}=0$
${t}_{1}=9$ ${x}_{1}=0.5$ $\text{Δ}{x}_{1}={x}_{1}-{x}_{0}=0.5$
${t}_{2}=18$ ${x}_{2}=0$ $\text{Δ}{x}_{2}={x}_{2}-{x}_{1}=-0.5$
${t}_{3}=33$ ${x}_{3}=1.0$ $\text{Δ}{x}_{3}={x}_{3}-{x}_{2}=1.0$
${t}_{4}=58$ ${x}_{4}=-0.75$ $\text{Δ}{x}_{4}={x}_{4}-{x}_{3}=-1.75$

## Solution

1. From the above table, the total displacement is
$\sum \text{Δ}{x}_{\text{i}}=0.5-0.5+1.0-1.75\phantom{\rule{0.2em}{0ex}}\text{km}=-0.75\phantom{\rule{0.2em}{0ex}}\text{km}\text{.}$
2. The magnitude of the total displacement is $|-0.75|\phantom{\rule{0.2em}{0ex}}\text{km}=0.75\phantom{\rule{0.2em}{0ex}}\text{km}$ .
3. $\text{Average velocity}=\frac{\text{Total}\phantom{\rule{0.2em}{0ex}}\text{displacement}}{\text{Elapsed}\phantom{\rule{0.2em}{0ex}}\text{time}}=\stackrel{\text{–}}{v}=\frac{-0.75\phantom{\rule{0.2em}{0ex}}\text{km}}{58\phantom{\rule{0.2em}{0ex}}\text{min}}=-0.013\phantom{\rule{0.2em}{0ex}}\text{km/min}$
4. The total distance traveled (sum of magnitudes of individual displacements) is ${x}_{\text{Total}}=\sum |\text{Δ}{x}_{\text{i}}|=0.5+0.5+1.0+1.75\phantom{\rule{0.2em}{0ex}}\text{km}=3.75\phantom{\rule{0.2em}{0ex}}\text{km}$ .
5. We can graph Jill’s position versus time as a useful aid to see the motion; the graph is shown in [link] . This graph depicts Jill’s position versus time. The average velocity is the slope of a line connecting the initial and final points.

## Significance

Jill’s total displacement is −0.75 km, which means at the end of her trip she ends up $0.75\phantom{\rule{0.2em}{0ex}}\text{km}$ due west of her home. The average velocity means if someone was to walk due west at $0.013$ km/min starting at the same time Jill left her home, they both would arrive at the final stopping point at the same time. Note that if Jill were to end her trip at her house, her total displacement would be zero, as well as her average velocity. The total distance traveled during the 58 minutes of elapsed time for her trip is 3.75 km.

Check Your Understanding A cyclist rides 3 km west and then turns around and rides 2 km east. (a) What is his displacement? (b) What is the distance traveled? (c) What is the magnitude of his displacement? (a) The rider’s displacement is $\text{Δ}x={x}_{\text{f}}-{x}_{0}=-1\phantom{\rule{0.2em}{0ex}}\text{km}$ . (The displacement is negative because we take east to be positive and west to be negative.) (b) The distance traveled is 3 km + 2 km = 5 km. (c) The magnitude of the displacement is 1 km.

## Summary

• Kinematics is the description of motion without considering its causes. In this chapter, it is limited to motion along a straight line, called one-dimensional motion.
• Displacement is the change in position of an object. The SI unit for displacement is the meter. Displacement has direction as well as magnitude.
• Distance traveled is the total length of the path traveled between two positions.
• Time is measured in terms of change. The time between two position points ${x}_{1}$ and ${x}_{2}$ is $\text{Δ}t={t}_{2}-{t}_{1}$ . Elapsed time for an event is $\text{Δ}t={t}_{\text{f}}-{t}_{0}$ , where ${t}_{\text{f}}$ is the final time and ${t}_{0}$ is the initial time. The initial time is often taken to be zero.
• Average velocity $\stackrel{\text{–}}{v}$ is defined as displacement divided by elapsed time. If ${x}_{1},{t}_{1}$ and ${x}_{2},{t}_{2}$ are two position time points, the average velocity between these points is
$\stackrel{\text{–}}{v}=\frac{\text{Δ}x}{\text{Δ}t}=\frac{{x}_{2}-{x}_{1}}{{t}_{2}-{t}_{1}}.$

## Conceptual questions

Give an example in which there are clear distinctions among distance traveled, displacement, and magnitude of displacement. Identify each quantity in your example specifically.

Under what circumstances does distance traveled equal magnitude of displacement? What is the only case in which magnitude of displacement and displacement are exactly the same?

Bacteria move back and forth using their flagella (structures that look like little tails). Speeds of up to 50 μm/s (50 × 10 −6 m/s) have been observed. The total distance traveled by a bacterium is large for its size, whereas its displacement is small. Why is this?

If the bacteria are moving back and forth, then the displacements are canceling each other and the final displacement is small.

Give an example of a device used to measure time and identify what change in that device indicates a change in time.

Does a car’s odometer measure distance traveled or displacement?

Distance traveled

During a given time interval the average velocity of an object is zero. What can you say conclude about its displacement over the time interval?

## Problems

Consider a coordinate system in which the positive x axis is directed upward vertically. What are the positions of a particle (a) 5.0 m directly above the origin and (b) 2.0 m below the origin?

A car is 2.0 km west of a traffic light at t = 0 and 5.0 km east of the light at t = 6.0 min. Assume the origin of the coordinate system is the light and the positive x direction is eastward. (a) What are the car’s position vectors at these two times? (b) What is the car’s displacement between 0 min and 6.0 min?

a. ${\stackrel{\to }{x}}_{1}=\left(-2.0\phantom{\rule{0.2em}{0ex}}\text{m}\right)\stackrel{^}{i}$ , ${\stackrel{\to }{x}}_{2}=\left(5.0\phantom{\rule{0.2em}{0ex}}\text{m}\right)\stackrel{^}{i}$ ; b. 7.0 m east

The Shanghai maglev train connects Longyang Road to Pudong International Airport, a distance of 30 km. The journey takes 8 minutes on average. What is the maglev train’s average velocity?

The position of a particle moving along the x -axis is given by $x\left(t\right)=4.0-2.0t$ m. (a) At what time does the particle cross the origin? (b) What is the displacement of the particle between $\text{t}=3.0\phantom{\rule{0.2em}{0ex}}\text{s}$ and $\text{t}=6.0\phantom{\rule{0.2em}{0ex}}\text{s}?$

a. $t=2.0$ s; b. $x\left(6.0\right)-x\left(3.0\right)=-8.0-\left(-2.0\right)=-6.0\phantom{\rule{0.2em}{0ex}}\text{m}$

A cyclist rides 8.0 km east for 20 minutes, then he turns and heads west for 8 minutes and 3.2 km. Finally, he rides east for 16 km, which takes 40 minutes. (a) What is the final displacement of the cyclist? (b) What is his average velocity?

On February 15, 2013, a superbolide meteor (brighter than the Sun) entered Earth’s atmosphere over Chelyabinsk, Russia, and exploded at an altitude of 23.5 km. Eyewitnesses could feel the intense heat from the fireball, and the blast wave from the explosion blew out windows in buildings. The blast wave took approximately 2 minutes 30 seconds to reach ground level. (a) What was the average velocity of the blast wave? b) Compare this with the speed of sound, which is 343 m/s at sea level.

a. 150.0 s, $\stackrel{\text{–}}{v}=156.7\phantom{\rule{0.2em}{0ex}}\text{m/s}$ ; b. 45.7% the speed of sound at sea level

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