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Welcome back to Physics 211PowerPoint Presentation

Welcome back to Physics 211

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### ConcepTest 8

### Adding vectors

### Projection of a vector

You are throwing a ball…

Vectors

- are used to denote quantities that have magnitude and direction
- can be added and subtracted
- can be multiplied or divided by a number
- can be manipulated graphically (i.e., by drawing them) or algebraically (usually by considering components)

Examples

- Numbers: temperature, pressure, volume ….
- Vectors: position, velocity, force

To add vector B to vector A:

Draw vector A.

Draw vector B with its tail starting from the tip of A.

The sum vector A+B is the vector drawn from the tail of vector A to the tip of vector B.

Two unknown vectors A and B are added.

The magnitude of the sum vector “A + B” (i.e., the quantity |A + B|)

1. is at least as great as |A| (i.e., the magnitude of A).

2. is at most as great as |A| + |B| (i.e., the magnitudes of A and B added).

3. must be equal to |A| + |B|.

4. can be greater than |A| + |B|.

Component of a vector

“How much a vector acts along some arbitrary direction”

Projection onto one of the coordinate axes (x, y, z)

Components

y

A=Ax+Ay

A

A=axi+ayj

Ay

q

i unit vector in x direction

x

Ax

j unit vector in y direction

ax,ay components of vector A

Projection of A along coordinate axes

More components

- Note:
ax=|A|cosq, ay=|A|sinq

- |A|2=ax2+ay2

Why components useful ?

- Addition: just add components
if C=A+B

cx=ax+bx; cy=ay+by

- Magnitude |A|2=ax2+ay2
- Direction tanq=ay/ax

A bird is flying along a straight line in a direction somewhere East of North. After the bird has flown a distance of 2.5 miles, it is 2 miles North of where it started.

How far to the East is it from its starting point?

1. 0 miles

2. 0.5 miles

3. 1.0 mile

4. 1.5 miles

2D Motion somewhere East of North. After the bird has flown a distance of 2.5 miles, it is 2 miles North of where it started.

Note:

component of position vector

along x direction is the x coordinate!

s – vector position

s=xi+yj

Describing motion with vectors somewhere East of North. After the bird has flown a distance of 2.5 miles, it is 2 miles North of where it started.

- Positions and displacements
s,Ds = sfinal - sinitial

- Velocities and changes in velocity:
vave= ––––,vinst= lim ––––,

Dv = vfinal - vinitial

- Acceleration: aave= ––––,ainst= lim ––––,

Dt0

Dt0

Quick Quiz 3.3 somewhere East of North. After the bird has flown a distance of 2.5 miles, it is 2 miles North of where it started.(from: Serway/Beichner, Physics)

Can any component of a vector ever be greater than the magnitude of the vector?

1. Yes

2. No

3. Not sure

Quick Quiz 3.4 somewhere East of North. After the bird has flown a distance of 2.5 miles, it is 2 miles North of where it started.(from: Serway/Beichner, Physics)

If one component of a vector is not zero, can the magnitude of the vector be zero?

1. Yes

2. No

3. Not sure

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