Physics
Per:
Name:
Distance and displacement
Date:
Adapted from “Physics by Inquiry” by the Physics Education Group, University of Washington
READ and RE-READ all your directions.
Quantitative descriptions of motion
Whenever an object moves along a straight line, it is possible to imagine a meterstick lying alongside
the path of the object. We can imagine this meterstick as extending forever in both directions. The
meterstick represents a coordinate system.
We can describe where an object is located by stating the number on the meterstick that is closest to a
particular point on the object. This number is called the position of the object. We use the symbol x to
represent position. In the figure above, the center of the square is next to the +2 cm mark, so it has
position +2 cm. Using the symbol x, we would say that x = 2 cm for the square. For the circle, x = -3
cm.
Displacement
If an object moves from an initial position xi to a final position xf, the number xf - xi is called the
displacement or the change of position of the object. We will use the symbol x for “xf – xi” (i.e.,
x = xf – xi), where x is read "delta x." In physics, we always use the Greek letter delta () to
mean “final value minus initial value”. Thus, the symbol  represents a (signed, positive or negative)
change in a quantity. The absolute value of any  quantity is called the magnitude (if x = -5 cm
then the “magnitude of x " = 5cm  | x | = 5cm).
A displacement is often represented by an arrow drawn from the initial position to the final position.
An arrow representing displacement is shown in the following figure.
xf
xi
Note that because we do most things graphically in this class, the physical length of the arrow is
important, so, just like shown above, the arrow’s tail must start right at the beginning point, and the
arrow’s tip (or head) must end right at the ending point.
xf
xi
xf
xi
Exercise 2.1
A. For each of the following, draw an arrow to represent each displacement as in the preceding
figure. Then calculate the displacement ( x ) for each of the motions. And state the magnitude
of x (| x |). In each case, xi is the initial position and xf is the final position.
xi
xf
1.
x =
2- -4 = 6
6
| x | =
xf
xi
2.
x =
-4 - 2 = -6
| x | =
6
| x | =
xf
4
| x | =
3
xf
xi
3.
x =
-2 - 6 = 4
xi
4.
x =
1 - 4 = -3
xf
xi
5.
x =
3 - -1 = 4
| x | =
4
B. What does the sign (+/-) of the displacement tell you about the motion? Explain.
The sign tells you the direction of the motion, the neg. to
the left and the pos to the right
C. What does the length of a displacement arrow represent? What does the direction the arrow is
pointing represent? Explain.
The length represents the magnitude of the displacement
and the arrow indicates the direction
Motion may begin at any position number. Often x = 0 is defined to be the place where the
motion begins, but this is not necessary.
We will frequently encounter situations in which the motion does not start at x = 0. The choice of
location for x = 0 is arbitrary; however, once the choice has been made for a particular problem, it
should not be changed. The location where the position is zero is often called the origin of the
coordinate system. The following exercise illustrates how the choice of the origin affects position
and displacement measurements.
Exercise 2.2
Two students conduct an experiment in which an object is moved from position xi to position xf.
Student 1 and student 2 each place a meterstick next to the line of motion, as shown below.
xi
xf
A. Answer the following questions about the motion.
Would either student draw different displacement arrow? Explain
No, both start and finish the arrows at the same physical
locations.
What value would student 1 read for xi?
-1 cm
What value would student 1 read for xf?
6 cm
What value would student 2 read for xi?
-6 cm
What value would student 2 read for xf?
1 cm
What value would student 1 calculate for x (show your work)?
6 - -1 = 7 cm
What value would student 2 calculate for x (show your work)?
1 - -6 = 7 cm
B. Student 3 is discussing displacements and coordinate systems with students 1 and 2. Consider
the following statement made by student 3:
"The sign and the magnitude of x never depend on the coordinate system. The displacement
will always be the same physical amount and direction”
Do you agree or disagree with student 3? If you agree, explain why. If you disagree, draw a
coordinate system that is incompatible with the statement made by student 3.
The measurement of Δx will be the same will be the
same as long as both people use the same scale (both
measuring in cm) and orientation of the axes (e.g. + #s
on right, - #s on left)
C. Summarize the results of this exercise by answering the questions below. Explain your
reasoning for each (refer back to exercises 2.2A and 2.2B).
Does the measurement of position depend on the choice of coordinate system?
The measurement of x does depend on the coordinate
system as both people measured different xi and xf
values.
Does the measurement of displacement depend on the choice of coordinate system?
The measurement of Δx typically does NOT depend on
the coordinate system as both people measured the
same Δx values (though if they had different
orientations or scales the values could have been
different).
Does the displacement arrow depend on the choice of coordinate system?
The displacement arrow never depends on the
coordinate system as both people will see the same
physical starting and ending positions and the arrow is
always drawn from the starting postion to the ending
position.
In this class, we will consider two displacements to be equal if they are characterized by arrows that
have identical magnitudes and directions.
Exercise 2.3
Are any of the displacements in Exercise 2.1 equal? If so, which ones? Explain.
#s 3 and 5 both have the same magnitude and directions,
so they are equivalent displacements
Exercise 2.4
The figure below shows a one-dimensional coordinate system. The location of a cricket is shown
at five different times. The cricket visited each point in alphabetical order (i.e.. the cricket started
at point P traveled to point Q, etc.).
R
S
P
Q
T
Centimeters
A. Fill in the following table with the distances, displacements, cumulative distances (all the
distances to that point added up), total displacements (often called the net displacement, and is
the displacement from the first point, point P, to the point in question, e.g. if an object started
at position -1cm and then the current position was at -9cm, the displacement would be -9 - -1 =
-8cm). Assume distance and displacement are both zero prior to point P. SHOW YOUR
WORK!
Segment Distance Traveled
During Segment
Displacement of
Cumulative
Total Displacement
Segment
Distance
(Δxnet)
PQ
3
3-0 = +3
3
+3
QR
7
-4 – 3 = -7
3+7=10
+3 – 7 = -4
RS
3
-1 - -4=+3
10+3=13
-4 + 3 = -1
ST
5.5
4.5 - -1=+5.5 13+5.5=18.5 -1+5.5= +4.5
B. For each pair of consecutive points, how is the distance traveled related to the displacement
between those two points (that is, is there a relationship between distance and displacement
between PQ, QR, etc.)?
The distance traveled is equal to the magnitude of the
displacement between consecutive points
Does the same relationship hold for points the entire trip from P to T (total distance to total
displacement)? Under what circumstances are the magnitudes of the displacement and the
distance traveled equal?
No, from P to T the distance is 18.5cm but the
displacement is only 4.5cm. So distance and magnitude
of displacement are only equal in a straight line path
with no direction change.
C. How is the net (total) displacement from point P to point T related mathematically to the
individual displacements between consecutive points (what would you have to do with each of
the individual displacements to end up equaling the total displacement)? Write an equation
that shows this relationship (Δxnet = ΔxPQ…).
ΔxPT = the sum of the individual displacements.
ΔxPT = ΔxPQ + ΔxQR + ΔxRS + ΔxST
Δxnet = ∑
What information does the net displacement give about the actual path that an object takes
between any two points (not necessarily consecutive points)? Explain.
Absolutely NOTHING!!! All the
displacement tells us is the distance and direction the
object ended up from its starting point, it tells us nothing
about how it got there, there are an infinite number of
paths it could have taken.
Displacements and other arrow can also be added graphically. In this module, we will use the
tip-to-tail, or sometimes called head-to-tail method to add arrows graphically. In this method,
the arrows are added by diagrammatically drawing one vector and at that vector’s head (point of
the arrow) place the tail (flat end) of the next arrow. The result of the addition (called the
resultant) is the net displacement, and is drawn as an arrow from the first tail to the second tip.
Tail
For example,
First vector
Head
+
=
Second arrow’s tail placed at the
first arrow’s tip
(Note the vertical separation is purely for ease of
viewing, otherwise vectors would overlap)
Net displacement (resultant) arrow is drawn from the tail of the first to the
tip of the last arrow. I usually show the resultant as dashed to distinguish it
Exercise 2.5
from the vectors that actually make up the motion
Draw each of the individual displacements from exercise 2.4 (PQ, QR, etc.) in the grid
below using one grid for one unit. Apply the head-to-tail method to graphically determine the net
displacement arrow (between points P and T). Start P at the dot shown.
Exercise 2.6
In the first part of its motion, an object moves to the right with a displacement represented by the
displacement arrow shown below.
x 1
A. Suppose that the object then completes the second part of its motion, yielding a net displacement
as shown in each of the following cases. For each of the cases, copy the original displacement arrow
and then draw a second displacement arrow representing x 2 that would yield the net displacement
arrow shown, such that →
→
→ . So, x 1 is the first vector you draw. Remember that the
net displacement would always be drawn from the start (tail) of the first vector to the tip of the
last vector (so you know where the first vector starts and where the last vector ends).
Δx1
Δx2
1) x net
Δx2
Δx1
2) x net
3) x net = 0Δx
2
Δx1
B. In which of the cases above does the object travel the greatest distance? Explain.
Case 1 because it started going to the right and then had to
backtrack and go further to the left.
C. State whether or not each of the following three cases is possible. If such a case is possible,
describe the circumstances under which it is possible. If it is not possible, explain why not.
1) The magnitude of the net displacement is greater than the total distance traveled
NOT POSSIBLE, an object cannot travel a total distance of
5m and end up 10m from where it started.
2) The magnitude of the net displacement is less than the total distance traveled.
Possible when the object changes direction and back tracks,
then the overall distance traveled is greater than the ending
distance from the starting point. (If you go 5m to the left and
then 5m to the right, you’ve traveled 10m but ended up 0m
from the starting pt)
3) The magnitude of the net displacement is equal to the total distance traveled.
Possible goes in a straight line without changing direction.
(if you go 5m to the right and then another 5m to the right
you’ll have gone 10m and will end up 10m from where you
started)