PreClass Notes: Chapter 3 - University of Toronto2015-06-22 1 © 2012 Pearson Education, Inc. Slide 1-1 PreClass Notes: Chapter 3 •From Essential University Physics 3rd Edition•by

2015-06-22

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© 2012 Pearson Education, Inc. Slide 1-1

PreClass Notes: Chapter 3

• From Essential University Physics 3rd Edition

• by Richard Wolfson, Middlebury College

• ©2016 by Pearson Education, Inc.

• Narration and extra little notes by Jason Harlow,

University of Toronto

• This video is meant for University of Toronto

students taking PHY131.


Outline

• 3.1 Vector Math

• 3.2 Velocity and Acceleration

in 2-dimensions

• 3.3 Relative Velocity

• 3.4 Constant Acceleration in x

and y

• 3.5 Projectile Motion

• 3.6 Motion on a Circular Path

“At what angle should this

penguin leave the water to

maximize the range of its

jump?” – R.Wolfson

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Vectors

A quantity that is fully

described by a single number is

called a scalar quantity (ie mass,

temperature, volume)

A quantity having both a

magnitude and a direction is

called a vector quantity

The geometric representation of

a vector is an arrow with the tail

of the arrow placed at the point

where the measurement is made

We label vectors by drawing a small arrow over the letter

that represents the vector, ie: r for position, v for velocity, a

for acceleration


Vector Addition

Suppose you walk 2.0 m in a direction is 30°from the x-axis.

(This might be 30°north of east, for example.)

Then you turn right a bit and walk 1.0 m parallel to x.

Your final position can be found as: 2 1r r r

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Vector Addition: Geometric Methods

It is often convenient to draw two vectors with their tails

together, as shown in (a) below

To evaluate F = D + E, you could move E over and use the

tip-to-tail rule, as shown in (b) below

Alternatively, F = D + E can be found as the diagonal of

the parallelogram defined by D and E, as shown in (c) below


The four segments are described by displacement vectors D1,

D2, D3 and D4

The hiker’s net displacement, an arrow from position 0 to 4, is

Vector addition is easily

extended to more than two vectors

The figure shows the path of a

hiker moving from initial position

0 to position 1, then 2, 3, and

finally arriving at position 4

The vector sum is found by using the tip-to-tail method three

times in succession

Addition of More than Two Vectors

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Vector Components


Vector Components

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Vector Math: Components Method

We can perform vector addition by adding the x- and y-

components separately

This method is called algebraic addition

For example, if D = A + B + C, then

Similarly, to find R = P – Q we would compute

To find T = cS, where c is a scalar, we would compute


Unit Vectors

Each vector in the figure to the

right has a magnitude of 1, no

units, and is parallel to a coordinate

axis

A vector with these properties is

called a unit vector

These unit vectors have the

special symbols

Unit vectors establish the directions of the positive axes of the

coordinate system

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Two-Dimensional Velocity

If the velocity vector’s angle

θ is measured from the positive

x-direction, the velocity

components are

where the particle’s speed is

Conversely, if we know the velocity components, we can

determine the direction of motion:


Two-Dimensional Acceleration

The figure to the right shows

the trajectory of a particle

moving in the x-y plane

The instantaneous velocity is

v1 at time t1 and v2 at a later

time t2

We can use vector

subtraction to find a during the

time interval Δt = t2 – t1

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Two-Dimensional Acceleration

The instantaneous acceleration is the limit average

acceleration as ∆t 0.

The instantaneous acceleration vector is shown along with the

instantaneous velocity in the figure.

By definition, is the rate at which is changing at that

instant.


Relative Motion

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Relative MotionIf we know an object’s velocity measured in one reference

frame, S, we can transform it into the velocity that

would be measured by an experimenter in a different

reference frame, S´, using the Galilean transformation of

velocity:

Or, in terms of components:


Relative Motion

• Example:

– A jetliner flies at 960 km/h relative

to the air, heading northward.

There’s a wind blowing eastward

at 190 km/h. In what direction

should the plane fly?

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Constant Acceleration: Big Idea If the acceleration is constant, then the two

components ax and ay are both constant

In this case, everything from Chapter 2 about constant-

acceleration kinematics applies to the components

The x-components and y-components of the motion can be

treated independently

They remain connected through the fact that t must be the

same for both

jaiaa yxˆˆ


Projectile Motion

Baseballs, tennis balls, Olympic divers, etc, all exhibit

projectile motion

A projectile is an object that moves in two dimensions

under the influence of only gravity

Projectile motion

extends the idea of free-

fall motion to include a

horizontal component of

velocity

Air resistance is

neglected

Projectiles in two dimensions follow a parabolic trajectory

as shown in the photoPhoto of “Curved water” by George Rex taken May 2010, Battersea Park, London Borough of Wandsworth, https://www.flickr.com/photos/36692623@N06/4640846739

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Projectile Motion

The start of a

projectile’s motion is

called the launch

The angle θ0 of the

initial velocity v0 above

the x-axis is called the

launch angle

where v0 is the initial speed

The initial velocity vector can be broken into components

𝑣𝑥0 = 𝑣0 cos 𝜃

𝑣𝑦0 = 𝑣0 sin 𝜃


Projectile Motion

Gravity acts downward

Therefore, a projectile

has no horizontal

acceleration

Thus

The vertical component of acceleration ay is −g of free fall

The horizontal component of ax is zero

Projectiles are in free fall

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Reasoning About Projectile Motion

If air resistance is neglected,

the balls hit the ground

simultaneously

The initial horizontal velocity

of the first ball has no influence

over its vertical motion

Neither ball has any initial

vertical motion, so both fall

distance h in the same amount of

time

A heavy ball is launched exactly horizontally at height h above

a horizontal field. At the exact instant that the ball is launched, a

second ball is simply dropped from height h. Which ball hits the

ground first?

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Range of a Projectile (Penguin Question)

This distance is sometimes

called the range of a projectile:

A projectile with initial speed v0 has a launch angle of θ above

the horizontal. How far does it travel over level ground before

it returns to the same elevation from which it was launched?

The maximum distance

occurs for θ = 45°

Trajectories of a projectile launched at

different angles with a speed of 50 m/s.


Circular Motion

Consider a ball on a roulette

wheel

It moves along a circular

path of radius r

Other examples of circular

motion are a satellite in an

orbit, or a ball on the end of a

string

Circular motion is an

example of two-dimensional

motion in a plane

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Uniform Circular Motion

To begin the study of circular

motion, consider a particle that

moves at constant speed

around a circle of radius r

This is called uniform

circular motion

The time interval to complete

one revolution is called the

period, T

The period T is related to the

speed v:


Centripetal

Acceleration

In uniform circular motion,

although the speed is constant,

there is an acceleration because

the direction of the velocity

vector is always changing

The acceleration of uniform

circular motion is called

centripetal acceleration

The direction of the centripetal acceleration is toward the

center of the circle

The magnitude of the centripetal acceleration is constant for

uniform circular motion

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Centripetal Acceleration

The figure shows the

velocity v1 at one instant and

the velocity v2 an

infinitesimal amount of time

dt later

By definition, a = dv/dt

By analyzing the isosceles

triangle of velocity vectors,

we can show that:


Nonuniform Circular Motion

• In nonuniform circular motion, speed and path

radius can both change.

• The acceleration has both radial and tangential

components:

r ta a a

• is perpendicular to while is tangential to .ra v ta v

2 2

t ra a a

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Nonuniform Circular Motion

PreClass Notes: Chapter 3 - University of Toronto2015-06-22 1 © 2012 Pearson Education, Inc. Slide 1-1 PreClass Notes: Chapter 3 •From Essential University Physics 3rd Edition•by

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