Physics 1710 Chapter 3 Vectors Demonstration: Egg Toss REVIEW

Physics 1710Physics 1710 Chapter 3 Vectors Chapter 3 Vectors

Demonstration:Demonstration:

Egg TossEgg Toss

REVIEWREVIEW

Why did the egg not break the first Why did the egg not break the first time it was caught but did the second time it was caught but did the second

time?time?

No Talking!No Talking!

Think!Think! Confer!Confer!

REVIEWREVIEW


Different accelerationDifferent acceleration

REVIEWREVIEW

a = (va = (vfinalfinal 22 – v – v initial initial 22)/ (2)/ (2∆x)∆x)

Why wear a seat belt or use air Why wear a seat belt or use air bags?bags?


Seat belt:Seat belt:

Air Bag Video


http://www.nhtsa.dot.gov/airbags/iihs3.html

11′′ Lecture: Lecture:

• A A VectorVector is a quantity that requires two or is a quantity that requires two or more numbers to define it and acts like the more numbers to define it and acts like the displacement vector.displacement vector.

• The The magnitude magnitude of a vector is the square root of of a vector is the square root of the sum of the squares of its the sum of the squares of its componentscomponents..

• A vector makes an A vector makes an angle to the angle to the xx-axis-axis whose whose tangent is equal to the ratio of thetangent is equal to the ratio of the y y-component to -component to the the xx-component.-component.


Is it far to Budapest?Is it far to Budapest?


Stranded Motorist asks Stranded Motorist asks horse cart driver, “Is it far horse cart driver, “Is it far to Budapest?”to Budapest?”

““Nem! It is not far.”Nem! It is not far.”


““Then, may I have a ride?”Then, may I have a ride?”

““Egan! Climb up.”Egan! Climb up.”

After a long time the After a long time the Motorist says, “I thought Motorist says, “I thought you said it was not far.”you said it was not far.”

The driver replies, “Oh! Now it The driver replies, “Oh! Now it is is veryvery far to Budapest.”far to Budapest.”

What is the problem?What is the problem?

The difference between The difference between distance and displacement.distance and displacement.

Where is the Student Union?Where is the Student Union?


Turn to your classmate and the one in the Turn to your classmate and the one in the odd numbered seat, tell the other where is odd numbered seat, tell the other where is

the Student Union.the Student Union.

Position is a Position is a vector.vector.

AA ScalarScalar is a entity that requires only one number is a entity that requires only one number to characterize it fully. (Like a scale.)to characterize it fully. (Like a scale.)

Examples:Examples:

What time is it?What time is it?

What is your weight?What is your weight?

What is the temperature of the room?What is the temperature of the room?

What is the weight of 100. Kg What is the weight of 100. Kg man?man? Weight = g m = 9.80 N/kg (100. kg) = Weight = g m = 9.80 N/kg (100. kg) = 980 N.980 N.


A A vectorvector is a quantity that requires more than is a quantity that requires more than one “component” to “tell the whole story.one “component” to “tell the whole story.

Example: Example:

Where is the treasure buried in the field?Where is the treasure buried in the field?

Use “orthogonal,” that is, perpendicular Use “orthogonal,” that is, perpendicular axes.axes.


Location in ManhattanLocation in Manhattan


22ndnd St and St and 44thth Ave Ave

44thth St and St and 22ndnd Ave Ave

(4,2)(4,2)

(2,4)(2,4)

Position in 2-Dimensions or higher is a Position in 2-Dimensions or higher is a VECTOR. VECTOR. We use We use boldface,boldface, not italic, to not italic, to denote a vector quantity, italics to denote the denote a vector quantity, italics to denote the scalar components.scalar components.

We often We often representrepresent a vector as a position on a a vector as a position on a graph with an arrow connecting the origin to the graph with an arrow connecting the origin to the position.position.


2-Dimensional Vector

Physics 1710Physics 1710Chapter 2 Motion in One Dimension—II Chapter 2 Motion in One Dimension—II

Y(m

)

X (m)

θθ

rr

Position Vector Position Vector rr

rr = = (x,y) = x (x,y) = x i + i + y y jj

x= r cos x= r cos θθ

y = r sin θy = r sin θ

| | r r | = | = r = r = √(x √(x 2 2 + y + y 22), ), θ = tan θ = tan –1–1(y/x)(y/x)

xx

yy

iijj

The length of the arrow represents the The length of the arrow represents the

magnitudemagnitude of the vector. In orthogonal of the vector. In orthogonal coordinates, the magnitude of vector coordinates, the magnitude of vector A A given given by:by:

∣∣A∣A∣ = √ [A = √ [Axx22 + A + Ayy

22 + A + Azz22 ] ]

80/20 Fact:80/20 Fact:


The The directiondirection of the vector of the vector AA is is characterized (two dimensions) by the characterized (two dimensions) by the angle angle it makes with the “x-axis.” it makes with the “x-axis.”

tan tan θθ = A= Ayy / A / Axx

80/20 Fact:80/20 Fact:


2-Dimensional Vector


Y(m

)

X (m)

rr

Position Vector Position Vector rr

| | r r | = | = r = r = √(x √(x 2 2 + y + y 22))

= = √(2.0 √(2.0 2 2 + 1.5 + 1.5 22))

= = √(4.0√(4.0 + 2.25 ) = √(6.25)+ 2.25 ) = √(6.25)

= 2.5 m= 2.5 m

xx

yy

One may combine vectors by “ One may combine vectors by “ vector vector additionaddition”:”:

C = C = AA + + BB

Then Then

C C xx= = AAxx+ B+ Bx x && C Cyy== AAyy+ B+ By y

Key point: Key point:

Add the components separately.Add the components separately.

Observe Observe strict segregation strict segregation of x and y parts. of x and y parts.

80/20 Fact:80/20 Fact:


The product of a scalar and a vector is a vector The product of a scalar and a vector is a vector for which for which every component is multiplied by every component is multiplied by the scalarthe scalar::

C = C = k k AA

CCx x = k A= k Axx

CCy y = k A= k Ayy

CCz z = k A= k Azz

80/20 Fact:80/20 Fact:


N.B. ( Note Well):N.B. ( Note Well):

ⅠⅠA + BA + BⅠ ≠ Ⅰ ≠ (A + B)(A + B)


Note:Note:

ⅠⅠAA + + BBⅠ = Ⅰ = √[(√[(AAxx+ B+ Bx x ) ) 22 + + ((AAyy+ B+ By y ) ) 22 ] ] ≤ (A + B)≤ (A + B)

Proof:Proof:

((AAxx+ B+ Bx x ) ) 22 + + ((AAyy+ B+ By y ) ) 22 ≤ (A+B)≤ (A+B)22 = A = A2 2 +2AB +B+2AB +B22

LHS = ALHS = Axx 22 +A +Ayy22 + B + Bx x

22 + B + By y 22 + + 22AAxxBBx x + + 22AAyyBBy y

RHS = ARHS = Axx22 + A + Ayy

22+ B+ Bx x 22 + B + By y

22 +2 +2√√(A(Axx2 2 BBxx

2 2 + A+ Ayy2 2 BByy

22 + A+ Ayy

2 2 BBxx2 2 + +

AAxx2 2 BByy

22) )

LHS LHS ≤ RHS≤ RHS

22AAxxBBx x + + 22AAyyBByy≤ ≤ 22√√(A(Axx2 2 BBxx


22 + A + Ayy2 2 BBxx

2 2 + A+ Axx2 2 BByy

22) )

AAx x 22BBx x

22+ 2 A+ 2 AxxBBx x AAyyBBy y ++ AAyy22

BBy y 22

≤ ≤ AAxx2 2 BBxx


22 + A + Ayy2 2 BBxx

2 2 + A+ Axx2 2

BByy22

2 A2 AxxBBx x AAyyBBy y ≤ ≤ AAyy2 2 BBxx

2 2 + A+ Axx2 2 BByy

2 2

iff iff 0 ≤ (0 ≤ (AAyyBBxx - A- Axx

BByy) ) 22


We often designate the components of the We often designate the components of the vector by vector by unit vectors ( i, j, k )unit vectors ( i, j, k ) the x,y, and z the x,y, and z components, respectively.components, respectively.

Thus, 2.0 Thus, 2.0 ii + 3.0 + 3.0 j j has an x-component of 2.0 has an x-component of 2.0 units and a y-component of 3.0 units. units and a y-component of 3.0 units.

Or (2.0, 3.0)Or (2.0, 3.0)

80/20 Fact:80/20 Fact:


Summary:Summary:

•To add vectors, simply add the components To add vectors, simply add the components separately.separately.

•Use the Pythagorean theorem for the magnitude.Use the Pythagorean theorem for the magnitude.

•Use trigonometry to get the angle.Use trigonometry to get the angle.

•The vector sum will always be equal or less than The vector sum will always be equal or less than the arithmetic sum of the magnitudes of the the arithmetic sum of the magnitudes of the vectors.vectors.


• The main point of today’s lecture.The main point of today’s lecture.

• A realization I had today.A realization I had today.

•A question I have. A question I have.

11′ Essay′ Essay::

One of the following:One of the following:



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Physics 1710 Chapter 3 Vectors Demonstration: Egg Toss REVIEW