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Position and Time

After understanding the role of the frames of reference, the next concepts to be introduced in the understanding of motion are position and time.

Positions are measured in units of length, such as meter (m), with respect to the origin of coordinates. In one dimension, to the right of the origin of coordinates, positions are positive; while to the left of origin of coordinates, positions are negative. The symbol for position is x.

What is the change in position for the car in the 4 seconds represented in the drawing? The answer is 70 m. What did you need to do to respond the previous question?

220 m -150 m =70 m

 In the animation above, the motion of the car is shown at two different times. The initial position of the car is identified in the drawing by the coordinate at the initial time , later  the car is at the coordinate called the  final position at the final time . The relative position between the initial and final positions are represented by the third drawing of the sequence . Based on the previous example, the change in position is calculated using the following relation . The change in position is the segment of the coordinate system represented in the last drawing of the sequence . In the same form, the change in time or time taken for this change to occur can be calculated using the following relation, .

 

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Distance and Displacement

The difference between distance traveled and displacement can be better understood when studied in a two dimensional frame of reference. The next applet shows a sequence of diagrams illustrating this ideas.

Suppose that, initially, there is a mobile at point A moving toward point B, Figure I . In this case, the mobile moves to point B following the path illustrated in Figure II . When the mobile is at point B, final position, how far is from the initial position, point A? To answer this question, it is necessary to measure the length of the straight line segment going from point A to point B, Figure III . The mobile followed the longer curvilinear path rather than the straight line path. Thus, two different measurements are derived from this analysis; first, the distance traveled  by the mobile which is defined as the length of the path followed by the mobile, Figure IV . The distance traveled can be measured by placing a string along the actual path of the mobile and then straightening the string. On the other side, in accordance with the definition for distance between two points, the length of the straight line (or shortest line) from the initial position, point A, to the final position, point B, is the displacement (later called the displacement vector), Figure V . Figure VI of the sequence shows both the distance traveled and the displacement . The last figure of the sequence, Figure VII, presents the distance traveled extended from a curvilinear line into a straight line. This last figure allows to compare the distance traveled with the displacement for the mobile .

The distance traveled and the displacement are measured in units of length. The symbol for distance traveled is d . The displacement is the same as the change in position.

Change in Position = Displacement

Notice that the displacement can be calculated after just to observations of the motion. The first observation results in information about the initial location of the mobile, initial position and the initial time. The second observation  results in information about the final position and time for the mobile. However, for the distance traveled, it is necessary to have a continuous number of position observations (unlimited number of observations) in order to be able to reproduce the actual path of the mobile. This is so, even when the time may only be measured twice. In science, it is usually expensive and difficult to do measurements. So even when, commonly,  people is more familiar with distance traveled, displacements are simpler to work with. In fact, just above, a relation is given for calculating displacements. There is not a simple relation for calculating distances.

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Introduction to Vectors

Distance and displacement have an additional difference, distances are scalar quantities while displacements are vector quantities. In this section, a brief definition of both will be given. For a more complete description of vectors go to Vectors.

In physics, in addition to the units of the quantities, quantities are divided in the two groups:

Physics Quantities

Scalar quantities are defined only by their magnitude. The magnitude is the numerical value of the quantity; example, the time of a class is 50 minutes. The magnitude is the number 50. The unit is the minutes. Remember that except for the case of counting objects, units are necessary for physics quantities. Scalar quantities are usually represented by italic type, t.

Time, Mass
Distance, Speed

Vector quantities are defined in terms of their magnitude and direction. The magnitude of a vector is the numerical value that defines the vector just as in the case of scalar quantities. The direction of a vector is as important as the magnitude when stating the characteristics of a vector. The direction of a vector on the surface of the Earth is usually defined in terms of the cardinal points (north, south, east, and west) or the xy-plane. Vertical directions are defined as up and down. Vector quantities are represented by bold type, x, and/or by .

Weight
Displacement, Velocity

In one dimension, the difference between vector quantities and scalar quantities are minimal. The only difference is associated with the sign of the quantity. For example, for the drawing at the top, 70 m is the distance traveled and also the displacement. However, for the drawing below, the distance travel still is 70 m but the displacement is -70 m. The negative sign provides the direction for the vector. Here, the convention used is that vectors pointing to the right are positive while vectors pointing to the left are negative. This convention will be use along these notes.

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Exercises

 
The smallest displacement is obtained when an object moves from

 

a)

A B   C

 

b)

A F

 

c)

A B   C D E F

N

d)

 A B   C D E F   A

 

e)

None of the above.

 

The smallest distance traveled is obtained when an object moves from

 

a)

A B   C

N

b)

A F

 

c)

A B   C D E F

 

d)

 A B   C D E F   A

 

e)

None of the above.

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by Luis F. Sez, Ph. D.    Comments and Suggestions: LSaez@dallaswinwin.com