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Chapter 2

Motion in One Dimension

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2.1 Kinematics Describes motion while ignoring the

agents that caused the motion For now, will consider motion in one

dimension Along a straight line

Will use the particle model A particle is a point-like object, has mass

but infinitesimal size

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Position Defined in terms of a

frame of reference One dimensional, so

generally the x- or y-axis

The object’s position is its location with respect to the frame of reference

Fig 2.1(a)

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Position-Time Graph The position-time

graph shows the motion of the particle (car)

The smooth curve is a guess as to what happened between the data points

Fig 2.1(b)

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Displacement Defined as the change in position during

some time interval Represented as x

SI units are meters (m) x can be positive or negative

Different than distance – the length of a path followed by a particle

if xxx

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Vectors and Scalars Vector quantities need both magnitude

(size or numerical value) and direction to completely describe them Will use + and – signs to indicate vector

directions Scalar quantities are completely

described by magnitude only

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Average Velocity The average velocity is rate at which

the displacement occurs

The dimensions are length / time [L/T] The SI units are m/s Is also the slope of the line in the

position – time graph

t

xx

t

xv if

average

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Average Velocity, cont Gives no details about the motion Gives the result of the motion It can be positive or negative

It depends on the sign of the displacement It can be interpreted graphically

It will be the slope of the position-time graph

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Displacement from a Velocity - Time Graph

The displacement of a particle during the time interval ti to tf is equal to the area under the curve between the initial and final points on a velocity-time curve

Fig 2.1(c)

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2.2 Instantaneous Velocity The limit of the average velocity as the

time interval becomes infinitesimally short, or as the time interval approaches zero

The instantaneous velocity indicates what is happening at every point of time

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Fig 2.2(a)

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Instantaneous Velocity, graph The instantaneous

velocity is the slope of the line tangent to the x vs. t curve

This would be the green line

The blue lines show that as t gets smaller, they approach the green line

Fig 2.2(b)

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Instantaneous Velocity, equations The general equation for instantaneous

velocity is

When “velocity” is used in the text, it will mean the instantaneous velocity

dtdx

tx

v lim0t

x

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Instantaneous Velocity, Signs

At point A, vx is positive The slope is positive

At point B, vx is zero The slope is zero

At point C, vx is negative The slope is negative

Fig 2.3

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Instantaneous Speed The instantaneous speed is the

magnitude of the instantaneous velocity vector Speed can never be negative

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Fig 2.4

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Fig 2.5

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2.3 Constant Velocity If the velocity of a particle is constant

Its instantaneous velocity at any instant is the same as the average velocity over a given time period

vx = v x avg = x / t

Also, xf = xi + vx t

These equations can be applied to particles or objects that can be modeled as a particle moving under constant velocity

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Fig 2.6

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2.4 Average Acceleration Acceleration is the rate of change of the

velocity

It is a measure of how rapidly the velocity is changing

Dimensions are L/T2

SI units are m/s2

tv

ttvv

a x

if

xixfavgx

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Instantaneous Acceleration The instantaneous acceleration is the

limit of the average acceleration as t approaches 0

2

2xx

0tx dtxd

dtdv

tv

lima

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Instantaneous Acceleration – Graph The slope of the velocity

vs. time graph is the acceleration

Positive values correspond to where velocity in the positive x direction is increasing

The acceleration is negative when the velocity is in the positive x direction and is decreasing

Fig 2.7

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Fig 2.8

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Fig 2.9

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Fig 2.10

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2.5 Negative Acceleration Negative acceleration does not

necessarily mean that an object is slowing down If the velocity is negative and the

acceleration is negative, the object is speeding up

Deceleration is commonly used to indicate an object is slowing down, but will not be used in this text

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Acceleration and Velocity, 1 When an object’s velocity and

acceleration are in the same direction, the object is speeding up in that direction

When an object’s velocity and acceleration are in the opposite direction, the object is slowing down

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Acceleration and Velocity, 2

The car is moving with constant positive velocity (shown by red arrows maintaining the same size)

Acceleration equals zero

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Acceleration and Velocity, 3

Velocity and acceleration are in the same direction Acceleration is uniform (blue arrows maintain the

same length) Velocity is increasing (red arrows are getting longer) This shows positive acceleration and positive velocity

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Acceleration and Velocity, 4

Acceleration and velocity are in opposite directions Acceleration is uniform (blue arrows maintain the

same length) Velocity is decreasing (red arrows are getting shorter) Positive velocity and negative acceleration

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Fig 2.11

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2.6 Kinematic Equations – Summary

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Fig 2.12

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Kinematic Equations The kinematic equations may be used

to solve any problem involving one-dimensional motion with a constant acceleration

You may need to use two of the equations to solve one problem

The equations are useful when you can model a situation as a particle under constant acceleration

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Kinematic Equations, specific For constant a, Can determine an object’s velocity at

any time t when we know its initial velocity and its acceleration

Does not give any information about displacement

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Kinematic Equations, specific For constant acceleration,

The average velocity can be expressed as the arithmetic mean of the initial and final velocities

Only valid when acceleration is constant

2vv

v xfxix

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Kinematic Equations, specific For constant acceleration,

This gives you the position of the particle in terms of time and velocities

Doesn’t give you the acceleration

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Kinematic Equations, specific For constant acceleration,

Gives final position in terms of velocity and acceleration

Doesn’t tell you about final velocity

2xxiif ta

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tvxx

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Kinematic Equations, specific For constant a,

Gives final velocity in terms of acceleration and displacement

Does not give any information about the time

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Graphical Look at Motion – displacement – time curve The slope of the

curve is the velocity The curved line

indicates the velocity is changing Therefore, there is

an acceleration

Fig 2.12(a)

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Graphical Look at Motion – velocity – time curve The slope gives the

acceleration The straight line

indicates a constant acceleration

Fig 2.12(b)

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The zero slope indicates a constant acceleration

Graphical Look at Motion – acceleration – time curve

Fig 2.12(c)

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Problem-Solving Strategy –Constant Acceleration Conceptualize

Establish a mental representation Categorize

Confirm there is a particle or an object that can be modeled as a particle

Confirm it is moving with a constant acceleration

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Problem-Solving Strategy –Constant Acceleration, cont Analyze

Set up the mathematical representation Choose the instant to call the initial time

and another to be the final time These are defined by the parts of the

problem of interest, not necessarily the actual beginning and ending of the motion

Choose the appropriate kinematic equation(s)

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Problem-Solving Strategy –Constant Acceleration, Final Finalize

Check to see if the answers are consistent with the mental and pictorial representations

Check to see if your results are realistic

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2.7 Freely Falling Objects A freely falling object is any object

moving freely under the influence of gravity alone

It does not depend upon the initial motion of the object Dropped – released from rest Thrown downward Thrown upward

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Galileo Galilei(1564-1642)

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Acceleration of Freely Falling Object The acceleration of an object in free fall is

directed downward, regardless of the initial motion

The magnitude of free fall acceleration is g = 9.80 m/s2

g decreases with increasing altitude g varies with latitude 9.80 m/s2 is the average at the earth’s surface

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Acceleration of Free Fall, cont. We will neglect air resistance Free fall motion is constantly

accelerated motion in one dimension Let upward be positive Use the kinematic equations with ay = g

= -9.80 m/s2

The negative sign indicates the direction of the acceleration is downward

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Fig 2.14

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Fig 2.15

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Free Fall Example Initial velocity at A is upward (+) and

acceleration is g (-9.8 m/s2) At B, the velocity is 0 and the

acceleration is g (-9.8 m/s2) At C, the velocity has the same

magnitude as at A, but is in the opposite direction

The displacement is –50.0 m (it ends up 50.0 m below its starting point)

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2.8

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