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Lecture 4: Vectors & Components

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Lecture 4: Vectors & Components

Questions of Yesterday

1)A skydiver jumps out of a hovering helicopter and a few seconds later a second skydiver jumps out so they both fall along the same vertical line relative to the helicopter.

1a) Does the difference in their velocities:

a) increase

b) decrease

c) stay the same

1b) What about the vertical distance between them?

2) I drop ball A and it hits the ground at t1. I throw ball B horizontally (v0y = 0) and it hits the ground at t2. Which is correct?

a) t1 < t2

b)t1 > t2

c) t1 = t2

y (m)

3

2

1

x (m)

-1

-3

-2

1

2

3

x (m)

-1

-2

-3

-2

-1

0

1

2

3

-3

Vector vs. Scalar Quantities

Vector Quantities: Magnitude and Direction

Ex. Displacement, Velocity, Acceleration

Scalar Quantities: Magnitude

Ex. Speed, Distance, Time, Mass

What about 2 Dimensions?

Vectors in 1 Dimension

Direction specified

solely by + or -

R

R

R

R

q

q

q

q

Vectors: Graphical Representation

Vector Quantities: Magnitude and Direction

Represent in 2D with arrow

Length of arrow = vector magnitude

Angle of arrow = vector direction

y (m)

R m at qoabove x-axis

3

Position of vector

not important

Vectors of equal

length & direction are equal

Can translate vectors for convenience (choose ref frame)

2

1

x (m)

-3

-2

-1

1

2

3

-1

-2

-3

B

B

A

A

A + B

Adding Vectors: Head-to-Tail

Must have same UNITS (true for scalars also)

Must add magnitudes AND directions..how?

A + B = ?

Head-to-Tail Method

B + A

B

B

A

A

Adding Vectors: Commutative Property

A + B = B + A ?

A + B

YES!

A + B = B + A

Can add vectors in any order

-A

B

A

A

-B

-B

A - B

Subtracting Vectors

A -> -A

Negative of vector = 180o rotation

A - B = A + (-B)

2A

A

A

-2A

v

x

Multiplying & Dividing Vectors by Scalars

2 * A = 2A

-2 * A = -2A

Ex. v = x/t

t = 3 s

N

E

W

S

lake B

190 km

30o

lake A

280 km

20o

Graphical Vector Techniques

1 box = 10 km

A plane flies from base camp to lake A a distance 280 km at a direction 20o north of east. After dropping off supplies, the plane flies to lake B, which is 190 km and 30.0o west of north from lake A.

Graphically determine the distance and direction from lake B to the base camp.

base camp

y

R

B

Ry

q

x

A

Rx

Rx = Rcosq

Ry = Rsinq

From magnitude (R) and direction (q) of R can determine Rx and Ry

Vector Components

Every vector can be described by its components

Component = projection of vector on x- or y-axis

y

= A + B

R

x

Ry

Vector Components

Can determine any vector from its components

y

- R2 = Rx2 + Ry2
- R = (Rx2 + Ry2)1/2
- tanq = Ry/Rx
- = tan-1(Ry/Rx)
-90 < q < 90

R

q

x

Rx

Vector Components

Can determine any vector from its components

Careful!

y

- R2 = Rx2 + Ry2
- R = (Rx2 + Ry2)1/2
- tanq = Ry/Rx
q = tan-1(Ry/Rx)

-90 < q < 90

(-x, +y)

(+x, +y)

II

I

x

III

IV

(-x, -y)

(+x, -y)

I, IV: q = tan-1(Ry/Rx)

II, III: q = tan-1(Ry/Rx) + 180o

Important to know direction of vector!

A

B

A + B

Vector Addition: Components

Why are components useful?

When is magnitude of A + B = A + B ?

- Rx = Ax + Bx + Cx….
- Ry = Ay + By + Cy….
q = tan-1(Ry/Rx)

-90 < q < 90

R = A + B + C…. = ?

Vector Addition: Components

lake B

Using components determine the distance and direction from lake B to the base camp.

190 km

30o

- Rx = Ax + Bx + Cx….
- Ry = Ay + By + Cy….
q = tan-1(Ry/Rx)

-90 < q < 90

lake A

280 km

20o

base camp

Vector Components: Problem #2

A man pushing a mop across a floor cause the mop to undergo two displacements. The first has a magnitude of 150 cm and makes an angle of 120o with the positive x-axis. The resultant displacement has a magnitude of 140 cm and is directed at an angle of 35.0o to the positive x-axis. Find the magnitude and direction of the second displacement.

Vector Components: Problem #3

An airplane starting from airport A flies 300 km east, then 350 km at 30.0o west of north, and then 150 km north to arrive finally at airport B. The next day, another plane flies directly from A to B in a straight line.

In what direction should the pilot travel in this direct flight?

How far will the pilot travel in the flight?

Questions of the Day

1) Can a vector A have a component greater than its magnitude A?

YES

b) NO

2) What are the signs of the x- and y-components

of A + B in this figure?

a) (x,y) = (+,+)

b) (+,-)

c) (-,+)

d) (-,-)

A

B