MFMcGrawChapter 3 - Revised: 2/1/20101 Motion in a Plane Chapter 3.

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Motion in a Plane

Chapter 3

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Motion in a Plane

• Vector Addition

• Velocity

• Acceleration

• Projectile motion

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Graphical Addition and Subtraction of Vectors

A vector is a quantity that has both a magnitude and a direction. Position is an example of a vector quantity.

A scalar is a quantity with no direction. The mass of an object is an example of a scalar quantity.

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Notation

Vector: FF

or

The magnitude of a vector: .or or FF

F

Scalar: m (not bold face; no arrow)

The direction of vector might be “35 south of east”; “20 above the +x-axis”; or….

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To add vectors graphically they must be placed “tip to tail”. The result (F1 + F2) points from the tail of the first vector to the tip of the second vector.

This is sometimes called the resultant vector R

F1

F2

R

Graphical Addition of Vectors

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Vector Simulation

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Examples

• Trig Table

• Vector Components

• Unit Vectors

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Think of vector subtraction A B as A+(B), where the vector B has the same magnitude as B but points in the opposite direction.

Graphical Subtraction of Vectors

Vectors may be moved any way you please (to place them tip to tail) provided that you do not change their length nor rotate them.

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Velocity

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y

x

ri rf

t

r

vav Points in the direction of r

r

vi

The instantaneous velocity points tangent to the path.vf

A particle moves along the blue path as shown. At time t1 its position is ri and at time t2 its position is rf.

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tt

rv lim

0

velocityousInstantane

The instantaneous velocity is represented by the slope of a line tangent to the curve on the graph of an object’s position versus time.

t

r

vav velocityAverage

t

xv x,av :be wouldcomponent - xThe

A displacement over an interval of time is a velocity

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Acceleration

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y

x

vi

ri rf

vf

A particle moves along the blue path as shown. At time t1 its position is r0 and at time t2 its position is rf.

v

Points in the direction of v.t

v

aav

The instantaneous acceleration can point in any direction.

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t

v

aavonaccelerati Average

A nonzero acceleration changes an object’s state of motion

tt

va lim

0

onaccelerati ousInstantane

These have interpretations similar to vav and v.

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Motion in a Plane with Constant Acceleration - Projectile

What is the motion of a struck baseball? Once it leaves the bat (if air resistance is negligible) only the force of gravity acts on the baseball.

Acceleration due to gravity has a constant value near the surface of the earth. We call it g = 9.8 m/s2

Only the vertical motion is affected by gravity

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The baseball has ax = 0 and ay = g, it moves with constant

velocity along the x-axis and with a changing velocity along the y-

axis.

Projectile Motion

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Example: An object is projected from the origin. The initial velocity components are vix = 7.07 m/s, and viy = 7.07 m/s.

Determine the x and y position of the object at 0.2 second intervals for 1.4 seconds. Also plot the results.

tvx

tatvy

ix

yiy

2

2

1

Since the object starts from the origin, y and x will represent the location of the object at time t.

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t (sec) x (meters) y (meters)

0 0 0

0.2 1.41 1.22

0.4 2.83 2.04

0.6 4.24 2.48

0.8 5.66 2.52

1.0 7.07 2.17

1.2 8.48 1.43

1.4 9.89 0.29

Example continued:

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0

2

4

6

8

10

12

0 0.5 1 1.5

t (sec)

x,y

(m

)

This is a plot of the x position (black points) and y position (red points) of the object as a function of time.

Example continued:

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Example continued:

0

0.5

1

1.5

2

2.5

3

0 2 4 6 8 10

x (m)

y (m

)

This is a plot of the y position versus x position for the object (its trajectory). The object’s path is a parabola.

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Example (text problem 3.50): An arrow is shot into the air with = 60° and vi = 20.0 m/s.

(a) What are vx and vy of the arrow when t = 3 sec?

The components of the initial velocity are:

m/s 3.17sin

m/s 0.10cos

iiy

iix

vv

vv

At t = 3 sec:m/s 1.12

m/s 0.10

tgvtavv

vtavv

iyyiyfy

ixxixfx

x

y

60°

vi

CONSTANT

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(b) What are the x and y components of the displacement of the arrow during the 3.0 sec interval?

y

x

r

m 80.72

1

2

1

m 0.3002

1

22

2

tgtvtatvyr

tvtatvxr

iyyiyy

ixxixx

Example continued:

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Example: How far does the arrow in the previous example land from where it is released?

The arrow lands when y = 0. 02

1 2 tgtvy iy

Solving for t: sec 53.32

g

vt iy

The distance traveled is:

m 35.302

1 2

tv

tatvx

ix

xix

What about the 2nd solution?

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Summary

• Adding and subtracting vectors (graphical method & component method)

• Velocity

• Acceleration

• Projectile motion (here ax = 0 and ay = g)

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Projectiles Examples

• Problem solving strategy

• Symmetry of the motion

• Contact forces versus long-range forces

• Dropped from a plane

• The home run