Position and Displacement in Mechanics

Kinematics begins by answering the simplest possible question about motion: where is the object? In mechanics the answer is called the object’s position. Motion is then described by how that position changes with time.

The existing PhysicsLibrary position entry gives the essential definition: position is the location of an object, commonly written as x in one dimension and treated as a vector in more than one dimension. This article develops that definition into the form needed for mechanics.

The central ideas are

position : r (t),

(1)

and

displacement : Δr = rf − ri.

(2)

Position depends on the chosen reference frame and coordinate origin. Displacement compares two positions and points directly from the initial position to the final position.

1 Position requires a reference

A statement such as “the particle is at 4 m” is incomplete unless a reference origin and direction have been specified. In one-dimensional motion we normally choose an x axis, an origin x = 0, and a positive direction. The position coordinate x then tells where the particle is relative to that origin.

In two or three dimensions, a position is conveniently represented by a position vector. In Cartesian coordinates,

r = xex + yey + zez.

(3)

The vector begins at the chosen origin and ends at the point occupied by the particle.

Figure 1. A position vector points from the chosen origin to the particle. Its Cartesian components are the coordinates of the point.

The SI unit of position is the meter, written m. Position can also be expressed in centimeters, kilometers, astronomical units, or any other appropriate unit of length.

2 Position in one dimension

For motion along a straight line, one signed coordinate is sufficient. We write

x = x(t).

(4)

A negative coordinate does not mean a negative distance. It means that the particle lies on the negative side of the chosen origin.

For example, the positions

x = − 3m

(5)

and

x = +3 m

(6)

are equally far from the origin but lie on opposite sides of it.

3 Position vectors in two and three dimensions

For planar motion,

r(t) = x (t)ex + y(t)ey.

(7)

For three-dimensional motion,

r(t) = x(t)ex + y(t)ey + z (t)ez.

(8)

The magnitude of the position vector is

∘ --2---2----2 r = |r| = x + y + z .

(9)

This magnitude is the straight-line distance from the chosen origin to the particle. It should not be confused with distance traveled along a trajectory.

Figure 2. In three dimensions, the position vector is reconstructed from its Cartesian components.

4 Position as a function of time

A moving particle occupies different positions at different times. Its position is therefore commonly written

r = r(t).

(10)

As t varies, the endpoint of r(t) traces the particle’s trajectory through physical space.

A trajectory is not the same thing as a position-time graph. A trajectory shows where the particle moves in space. A one-dimensional position-time graph plots a coordinate such as x(t) vertically against time horizontally.

5 Displacement is change in position

Suppose a particle moves from an initial position r_i to a final position r_f. Its displacement is defined by

Δr = rf − ri.

(11)

The Greek capital letter Δ means a finite change. In one dimension,

Δx = xf − xi.

(12)

In Cartesian three-dimensional form,

Δr = (xf − xi)ex + (yf − yi)ey + (zf − zi)ez.

(13)

Thus displacement is obtained by subtracting the initial position vector from the final position vector component by component.

6 Displacement depends only on the endpoints

The displacement vector connects the initial and final points directly. It does not record the path followed between them.

Figure 3. The curved trajectory represents the path traveled. The displacement Δr is the direct vector from the initial point to the final point.

If the motion passes through intermediate positions r₁,r₂,…, then the individual displacements add:

(r1 − r0) + (r2 − r1) + ⋅ ⋅⋅ + (rf − rn) = rf − r0.

(14)

The intermediate position vectors cancel. This is the vector form of the familiar “final minus initial” rule.

7 Displacement is not distance traveled

The distance traveled is the total length of the path followed by the particle. It is a nonnegative scalar. Displacement is a vector determined only by the endpoints.

For any motion between two points,

distance traveled ≥ |Δr |.

(15)

If a particle returns to its starting point, then

Δr = 0,

(16)

although the distance traveled may be large.

This difference is fundamental. A runner who completes one full lap of a circular track has nonzero distance traveled but zero total displacement.

8 Position-time graphs

For one-dimensional motion, a graph of x versus t records the coordinate position at each time.

Figure 4. On a one-dimensional position-time graph, the displacement between t₁ and t₂ is Δx = x(t₂) − x(t₁). The graph is not a drawing of the physical path.

x(t1) = 2m

(17)

and

x(t ) = − 1 m, 2

(18)

then

Δx = − 1 − 2 = − 3 m.

(19)

The negative sign identifies the direction of the net change relative to the chosen positive axis.

9 Changing the coordinate origin

Position depends on where the origin is placed. Suppose a new Cartesian origin is shifted by a constant vector a while the axes keep the same orientation. The new position vector is

′ r = r − a.

(20)

For two positions,

′ Δr = (rf − a) − (ri − a ).

(21)

Therefore

′ Δr = rf − ri = Δr.

(22)

A constant shift of the coordinate origin changes the numerical position vectors but not the physical displacement between the same two points.

This statement concerns a fixed coordinate translation. If two reference frames move relative to one another during the time interval, their measured displacements can differ, as developed in M00-06.

10 Differential displacement: a preview

For a small change of position in fixed Cartesian coordinates,

dr = dx e + dye + dz e . x y z

(23)

The finite displacement Δr compares two separated positions. The differential dr describes an infinitesimal change along a trajectory. In the next kinematics articles, dividing changes of position by changes of time leads to velocity.

11 Worked example 1: signed position, displacement, and distance

A cart begins at

x0 = − 2 m,

(24)

moves to

x1 = 4 m,

(25)

and then moves back to

x2 = 1 m.

(26)

The first displacement is

Δx1 = 4 − (− 2) = 6m.

(27)

The second displacement is

Δx = 1 − 4 = − 3 m. 2

(28)

The total displacement is

Δx = 1 − (− 2 ) = 3 m.

(29)

The total distance traveled is

6 + 3 = 9m.

(30)

Thus

displacement = +3 m,

(31)

while

distance = 9 m.

(32)

The difference occurs because the cart reversed direction.

12 Worked example 2: displacement in three dimensions

A particle moves from

ri = 2ex − ey + 3ez m

(33)

rf = − ex + 3ey + 7ez m.

(34)

The displacement is

Δr = rf − ri.

(35)

Subtracting components gives

Δr = − 3ex + 4ey + 4ez m.

(36)

Its magnitude is

∘ ---------------- √ --- |Δr | = (− 3)2 + 42 + 42 = 41 m ≈ 6.40 m.

(37)

This is the straight-line separation of the endpoints, regardless of the actual path traveled between them.

13 Worked example 3: quarter-circle path

A particle moves counterclockwise along a quarter circle of radius R = 2.0 m from the point (R, 0) to (0,R).

The path length is one quarter of the circumference:

1- πR-- s = 4(2πR ) = 2 .

(38)

For R = 2.0 m,

s = π m ≈ 3.14 m.

(39)

The displacement vector is

Δr = (0 − R )ex + (R − 0)ey,

(40)

Δr = − Rex + Rey.

(41)

Its magnitude is

√ -- √ -- |Δr | = R 2 = 2 2m ≈ 2.83 m.

(42)

The traveled distance and displacement magnitude are different because the particle followed a curved path.

14 Common mistakes

Confusing position with distance from the origin. Position in one dimension is signed, and in multiple dimensions it is a vector.
Using initial minus final. Displacement is final position minus initial position.
Adding path lengths to obtain displacement. Path length gives distance traveled, not displacement.
Subtracting vector magnitudes instead of vectors. In two or three dimensions, compute r_f − r_i component by component.
Treating a position-time graph as the physical trajectory. The axes of such a graph are position coordinate and time, not two spatial coordinates.
Assuming position is absolute. Numerical position depends on the chosen origin and reference frame.

15 Practice problems

M01-01-P01

A particle is at Cartesian coordinates (3,−4) m. Write its position vector and find its magnitude.

M01-01-P02

A cart moves from x_i = 7 m to x_f = −2 m. Find its displacement.

M01-01-P03

A person walks from x = 0 to x = 8 m, then back to x = 3 m. Find the total displacement and total distance traveled.

M01-01-P04

A particle moves from (1, 2) m to (7,−6) m. Find the displacement vector and its magnitude.

M01-01-P05

A particle moves from (2,−1, 4) m to (−3, 5, 1) m. Find Δr and |Δr|.

M01-01-P06

A runner completes one full lap of a circular track of radius 50 m. Find the total displacement and total distance traveled.

M01-01-P07

A particle moves along a semicircle of radius R from one end of a diameter to the other. Express the distance traveled and displacement magnitude in terms of R.

M01-01-P08

The positions of a particle at three successive instants are r₀, r₁, and r₂. Show algebraically that Δr₀₁ + Δr₁₂ = Δr₀₂.

M01-01-P09

A one-dimensional position is given by

x(t) = 2 + 3t − t2 m,

(43)

with t in seconds. Find the positions at t = 1 s and t = 4 s and the displacement over that interval.

M01-01-P10

A coordinate origin is shifted by a = 5e_x − 2e_y m. An object moves from r_i = 8e_x + e_y m to r_f = 11e_x + 7e_y m. Find the initial and final position vectors in the shifted coordinates and verify that the displacement is unchanged.

16 Practice answer check

r = 3e_x − 4e_y m, |r| = 5 m.
Δx = −9 m.
Displacement = +3 m; distance = 13 m.
Δr = 6e_x − 8e_y m; magnitude 10 m.
Δr = −5e_x + 6e_y − 3e_z m; magnitude m ≈ 8.37 m.
Displacement = 0; distance = 100π m.
Distance = πR; displacement magnitude = 2R.
The r₁ terms cancel, leaving r₂ − r₀.
x(1) = 4 m, x(4) = −2 m, displacement = −6 m.
r_i′ = 3e_x + 3e_y m, r_f′ = 6e_x + 9e_y m, and Δr = 3e_x + 6e_y m in either origin.

17 Summary

The essential results are:

Position specifies where an object is relative to a chosen frame and origin.
In three-dimensional Cartesian coordinates,
$r = xex + yey + zez.$ (44)
Displacement is the change in position,
$Δr = rf − ri.$ (45)
Displacement depends only on the endpoints; distance traveled depends on the path.
For any path,
$distance traveled ≥ |Δr |.$ (46)
A closed trip has zero total displacement even when the distance traveled is nonzero.
A constant shift of coordinate origin changes position vectors but not the displacement between the same two points.

The next article, M01-02, develops velocity from the rate of change of position.

Source and provenance note

This entry upgrades the existing PhysicsLibrary position definition (object 251). The exposition, examples, figures, and exercise set are original PhysicsLibrary material. Scope and terminology were benchmarked against the archived 2016 CC BY 4.0 University Physics volume 1 treatment of position/displacement and the CC BY-SA 4.0 UCD Physics 9A Classical Mechanics treatment of one- and multi-dimensional motion.

References

[1] PhysicsLibrary, position, object 251, existing definition entry.

[2] OpenStax / cnxuniphysics, University Physics Volume 1, archived 2016 BCcampus clone, sections on position, displacement, and displacement vectors, CC BY 4.0.

[3] T. Weideman, UCD Physics 9A - Classical Mechanics, sections 1.3 and 1.6, Physics LibreTexts, CC BY-SA 4.0.

[4] J. R. Taylor, Classical Mechanics, University Science Books, 2005.

"Position and Displacement in Mechanics" is owned by bloftin.
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