Chapter 3 : Motion

Weerachai Siripunvaraporn

Department of Physics, Faculty of ScienceMahidol University

email&msn : wsiripun2004@hotmail.com

Types of Motion

Translational An example is a car traveling on a highway.

Circular and Rotational An example is the Earth’s spin on its axis.

× Vibrational An example is the back-and-forth movement

of a pendulum.

Introduction

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What is in this chapter ?

1-D Motion : Horizontal

1-D : Vertical

2-D motion : circular

2-D : projectile2-D & 3-D motion

Particle Model & Terms used to describe motionWe will use the particle model.

A particle is a point-like object; has mass but infinitesimal size

PositionDistance & DisplacementSpeed & Velocity

Average & InstantaneousAcceleration

Average & Instantaneous

PositionThe object’s position is its location with respect to a chosen reference point.

Consider the point to be the origin of a coordinate system.

Only interested in the car’s translational motion, so model as a particle

Section 2.1

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You need a reference point.

You need coordinate system.

Position

Position: Where something is located.

You need a reference point.

You need coordinate system.

How can I define a reference point and a coordinate system?

Answer: Anything you like!

Representations of the Motion of CarVarious representations include:

Pictorial Graphical Tabular Mathematical

The goal in many problems

Section 2.1

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Position-Time GraphThe position-time graph shows the motion of the particle (car).The smooth curve is a guess as to what happened between the data points.

Section 2.1

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Data TableThe table gives the actual data collected during the motion of the object (car).Positive is defined as being to the right.

Section 2.1

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Displacement & Distance : Change in position

Displacement is defined as the change in position during some time interval.

Represented as x

x ≡ xf - xi

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

Different than distance Distance is the length of a path followed by a

particle.

Section 2.1

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Displacement & Distance : Examples

• Calculate displacement & distance from A to B?

• Calculate displacement & distance from A to C?

• Calculate displacement & distance from A to F?

What do you learn?

Distance vs. Displacement – An Example

Assume a player moves from one end of the court to the other and back. Distance is twice the length of the court

Distance is always positiveDisplacement is zero

Δx = xf – xi = 0 since xf = xi

Section 2.1

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Average Velocity(how fast the object is moving with direction)

The average velocity is rate at which the displacement occurs.

The x indicates motion along the x-axis.The dimensions are length / time [L/T]The SI units are m/sIs also the slope of the line in the position – time graph

Section 2.1

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Average Speed(how fast the object is moving)

Speed is a scalar quantity. Has the same units as velocity Defined as total distance / total time:

The speed has no direction and is always expressed as a positive number.Neither average velocity nor average speed gives details about the trip described.The SI units are m/s

Section 2.1

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Average velocity & Average speed: Examples

• Calculate average velocity & average speed from A to B?

• Calculate average velocity & average speed from A to C?

• Calculate average velocity & average speed from A to F?

What do you learn?

Average Speed and Average Velocity

The average speed is not the magnitude of the average velocity.

For example, a runner ends at her starting point. Her displacement is zero. Therefore, her velocity is zero. However, the distance traveled is not zero, so the

speed is not zero.

Section 2.1

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Instantaneous Velocity The limit of the average velocity as the time interval becomes infinitesimally short, or as the time interval approaches zero , i.e. t → 0, .The instantaneous velocity indicates what is happening at every point of time.

The instantaneous speed of a particle is defined as the magnitude of its instantaneous velocity, and no direction. For example, if one particle has an instantaneous velocity of +25 m/s along a given line and another particle has an instantaneous velocity of -25 m/s along the same line, both have a speed of 25 m/s.

“Velocity” and “speed” will indicate instantaneous values.Average will be used when the average velocity or average speed is indicated.

Acceleration

average

instantaneous

Motion Diagrams

A motion diagram can be formed by imagining the stroboscope photograph of a moving object.Red arrows represent velocity.Purple arrows represent acceleration.

Section 2.5

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What do you learn?

Acceleration and Velocity, DirectionsWhen an object’s velocity and acceleration are in the same direction, the object is speeding up.When an object’s velocity and acceleration are in the opposite direction, the object is slowing down.

Section 2.4

Negative acceleration does not necessarily mean the object is slowing down.

If the acceleration and velocity are both negative, the object is speeding up.

The word deceleration has the connotation of slowing down. This word will not be used in the text or in Physics.

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Kinematic Equations

The kinematic equations can be used with any particle under uniform acceleration.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.Many times there is more than one way to solve a problem.

Section 2.6CH2

Kinematic Equations, 1

For constant ax,

Can determine an object’s velocity at any time t when we know its initial velocity and its acceleration

Assumes ti = 0 and tf = t

Does not give any information about displacement

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Kinematic Equations, 2For constant acceleration,

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

This applies only in situations where the acceleration is constant.

Section 2.6CH2

Kinematic Equations, 3For constant acceleration,

This gives you the position of the particle in terms of time and velocities.Doesn’t give you the acceleration

Section 2.6CH2

Kinematic Equations, 4

For constant acceleration,

Gives final position in terms of velocity and accelerationDoesn’t tell you about final velocity

Section 2.6CH2

Kinematic Equations, 5For constant a,

Gives final velocity in terms of acceleration and displacementDoes not give any information about the time

Section 2.6CH2

When a = 0

When the acceleration is zero, vxf = vxi = vx

xf = xi + vx t

The constant acceleration model reduces to the constant velocity model.

Section 2.6CH2

Kinematic Equations – summary

Section 2.6CH2

Problem-Solving

Understand the problem, make a list of what is given or what can be inferred from the problem.

Identify what is asked for in the problem (i.e. identify the unknown)

Examine the problem to determine which physical principles or equations are involved.

Select the coordinate system and reference point. Substituting the given and inferred information into

the equations with their units and then solve equations algebraically for the unknown. (Make sure your Math is correct!)

Check the answer to see if it is reasonable.

1-D Motion: Horizontal & Vertical

Horizontal motion : x-axis

Vertical motion : y-axis (or z-axis)

A freely falling object is any object moving freely under the influence of gravity alone, regardless of its initial motion. Objects thrown upward or downward and those released from rest are all falling freely once they are released. Any freely falling object experiences an acceleration directed downward, regardles

s of its initial motion.

What is the acceleration of a ball and its direction?

g = 10 m/s2

Direction :

f iv = v +at

f i f i1

x = x + v + v t2

2f i i

1x = x + v t + at

2

2 2f i f iv = v +2a x - x

f iv = v +at

f i f i1

y = y + v + v t2

2f i i

1y = y + v t + at

2

2 2f i f iv = v +2a y - y

a is g in vertical direction, but sign will be determined by coordinate system.

X-Dir : motion Y-Dir : motion

Before (1-D) & Now (2-D)

2-D motion

Terms used to describe 2-D motion

PositionDistance & DisplacementSpeed & Velocity

Average & InstantaneousAcceleration

Average & Instantaneous

Position and Displacement

The position of an object is described by its position vector,The displacement of the object is defined as the change in its position.

r

f ir r r

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In two- or three-dimensional kinematics, everything is the same as in one-dimensional motion except that we must now use full vector notation.

Positive and negative signs are no longer sufficient to determine the direction.

Velocity and Acceleration

The magnitude of the instantaneous velocity vector is the speed.

Producing An Acceleration

Various changes in a particle’s motion may produce an acceleration.

The magnitude of the velocity vector may change. The direction of the velocity vector may change.

Even if the magnitude remains constant Both may change simultaneously

Section 4.1

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Problem Solving: same as 1-D motion

But in some case, we may have to divide into x- and y- directions and consider separately.

x

y

If Earth doesn’t have gravity, what would happen to the ball?

Projectile is a motion of a particle in a curved path.

x

y

But Earth DOES have gravity, what would happen to the ball?

g

Two assumptions:(1) the free-fall acceleration g is constant over the range of

motion and is directed downward,

(2) the effect of air resistance is negligible

Projectile is a motion of a particle in a curved path.

Projectile motion simultaneously combines motion in horizontal and vertical directions together.

We can then divide Projectile motion into motion in x-dir and y-dir and consider them separately.

Fact:

Initial velocity:

vx = v cos and vy = v sin

Acceleration :

ax = 0 and ay = -g

Time : t is total motion time.

(equal time in both directions)

Time is scalar, no direction.

Fact:

Initial velocity:

vx = v cos and vy = v sin

Acceleration :

ax = 0 and ay = -g

Time : t is total motion time. (equal time in both directions)

You are driving at 100 km/hr. This sentence is actually not completed.

The first rule in physics is “defining your reference”. What’s the reference here?

It’s the Earth.

i.e. you’re driving at a 100 km/hr relative to the Earth.

If another car goes at the same speed and direction, what would be its velocity relative to you? What would be yours relative to that car?

If another car goes at the same speed but to the opposite direction, what would be its velocity relative to you? What would be yours?

You can easily answer that by experience, here we will show you how to answer that using Physics!