General Physics I

Motion in One Dimension

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Position and Velocity

Spring 2011

Position

• To simplify the study of motion we consider point

particles, i.e., objects that can be represented as a

single point in space. In many cases, real objects

behave as if all their mass is concentrated at one

point, so the point-particle simplification is

reasonable.

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reasonable.

• Motion of a particle occurs when the position of the

particle changes. In one-dimensional (straight-line)

motion, position is specified by a location along the

x (or y) axis, which is taken to be the line along

which the particle moves.

Position

4 mix=+5 mix=−

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4 mix=+5 mix=−

Displacement• Displacement is a difference or change in position.

.if

x x x∆ = −

• The magnitude (absolute

value) tells you the straight-

line distance between the

two points. The sign of the

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two points. The sign of the

displacement tells you the

direction of the motion:

Positive means motion in

the positive x direction

(right); negative means

motion in the negative x

direction (left).

Distance

• If a particle travels 4 meters from A to B and back

to A, the displacement is zero. However, the

distance is not zero. The distance is the total path

length, which is 8 meters for the motion described

above.

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4 meters

B

Average Speed and Average Velocity• Consider a particle that moves between point A and point

B. The average speed of the particle is defined as

A B

xi xf

distanceAverage speed = = .time interval

dt∆

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,displacementtime interval

.ifx av

if

x xxv

t tt=

−∆= =

−∆

xi xf

• Note that the average speed is always positive because the

time interval is always positive and the distance is always

positive (unless the particle did not move at all).

• The average velocity of the particle is defined as

A particle moves 5.8 m along a straight line from point A to point B. The particle then moves back to A. The entire trip takes 5.2 s. Which of the following is true?

1. The average speed is zero and the average velocity is 1.1 m/s.

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2. The average velocity is zero and the average speed is 1.1 m/s.

3. The average velocity is zero and the average speed is 2.2 m/s.

4. The average speed is zero and the average velocity is 2.2 m/s.

Instantaneous Velocity• The average velocity has the same sign, i.e., direction, as the

displacement because the time interval is always positive.

• Note that the average velocity over a finite time interval tells

you nothing about how fast the particle is traveling at any

single point in time (speedometer reading) or the direction at

that instant. The velocity at a single instant in time is called the instantaneous velocity (v ). The instantaneous velocity is

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the instantaneous velocity (vx). The instantaneous velocity is

the average velocity when the time interval is reduced to an

infinitesimally small size. The instantaneous speed is the

magnitude (absolute value) of the instantaneous velocity. The

instantaneous speed is the reading a speedometer gives.

• Like the average velocity, the instantaneous velocity has a

magnitude (size) and a direction (its sign). They are

examples of vector quantities.

Position-Time Graphs

• A position-time (x-t) graph

gives the position of a particle

at every instant in time during

a given time interval.

• The average velocity is the

slope of a straight line drawn

between points representing

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between points representing

the initial and final states of

motion.

• The instantaneous velocity at

an instant in time is the slope

of the tangent to the x-t graph

at that instant.

Relationship Between Position-Time

Graph and the Motion

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A B C

Motion Diagram

DE C

Constructing the Velocity-Time Graph

from a Position-Time Graph

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Velocity

.xxvt

∆=∆

(constant velocity only)

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A person initially at point P in the illustration stays there

a while and then moves along the axis to Q and stays there a

while. She then runs quickly to R, stays there a while, and

then strolls slowly back to P. Which of the position vs. time graphs

below correctly represents this motion?

1. Figure 1.

2. Figure 2.

3. Figure 3.

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3. Figure 3.

4. Figure 4.

5. Figure 5.

6. Figure 6.

Workbook: Chapter 2, Questions 4, 6, 10(c)

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