Chapter 2

Motion in One Dimension

Kinematics To describe motion while ignoring the

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

dimension - along a straight line -, and use the particle model. A particle is a point-like object, has mass

but infinitesimal size

Position Defined in terms of

a frame of reference The reference frame

must has an origin. The object’s position

is its location with respect to the origin of the frame of reference.

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

Displacement Defined as the change in position during

some time interval t Represented as x

Xi is the initial position and Xf is the final position.

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

if xxx

Average Velocity The average velocity is rate at which

the displacement occurs

The dimensions are length / time [L/T] The slope of the line connecting the

initial and final points in the position – time graph

t

xx

t

xv if

average

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

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

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

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

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

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

Instantaneous Speed The instantaneous speed is the

magnitude of the instantaneous velocity vector Speed can never be negative

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

2

0lim

dt

xd

dt

dx

dt

d

dt

dv

t

va xx

tx

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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 2.10

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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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2.6 Motions 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

tvtvv

tvvtvx

avgxxixf

xixfxi

,)(2

1

)(2

1

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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 formula gives the position of the particle in terms of the initial and final velocities

The acceleration is not given.

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Kinematic Equations, specific As constant acceleration is given,

Gives final position in terms of initial velocity and acceleration

The final velocity is not necessary in this formula.

2xxiif ta

21

tvxx

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Kinematic Equations, specific For constant accelation,

The final velocity is determined by initial velocity, acceleration and displacement

There is no information about the time interval of the traveling.

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Kinematic Equations – Summary

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The zero slope indicates a constant acceleration

Graphical Look of Motion – acceleration – time curve

Fig 2.12(c)

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

acceleration The straight line

indicates a constant acceleration

Fig 2.12(b)

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Graphical Look of 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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Galileo Galilei(1564-1642)

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Fig 2.14

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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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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 2.15

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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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Exercises of chapter 2

4, 8, 14, 24, 28, 32, 33, 41, 44, 55, 60