Vectors and Relative Motion

Vector QuantityFully described by both

magnitude (number plus units) AND direction

Represented by arrows-velocity-acceleration-force

Scalar Quantity

Fully described by magnitude (number plus units) alone

-mass-temperature

Adding VectorsVectors in one dimension are added algebraically:

3 m, North + 4 m, North = 7 m, North3 m + 4 m = 7 m

3 m, North + 4 m, South = 1 m, South

For a vector-- Sign does not represent value, it represents direction!

Traditionally: Up/Right (+) Down/Left (-)

3 m + (-4 m) = -1 m

Adding Vectors in 2 Dimensions- Vectors add Trigonometrically Using Head to Tail Method:

3.0 m

4.0 m

6.5 m

3.0 m 4.0 m

2.2 m

3.0 m + 4.0 m = 6.5 m 3.0 m + 4.0 m = 2.2 m

N

Vector diagrams show magnitude and direction of vectors and their resultant!

8.0 N + 6.0 N = ? 2.0 N ≤ ? ≤ 14 N

Notice Vector Direction: In relation to + x axis

Vector Direction: By agreement, vectors are generally described by how many degrees the vector is rotated from the + x axis

30˚30˚ 150˚

Negative 2D vectors:

A

- A

180˚ opposite

Resolution (Decomposition) of Vectors

If you move a box 8.0 m @ 30.0˚ from O:

30.0˚

8.0 mBy Geometry:

4.0 m

6.9 m

The box has moved– 6.9 m to the right ( +x)

These values would be the components of the given vector !

Ø =30.0˚

d = 8.0 mdy

dx

sinø =dy

d

dy = sinø(d) = sin30.0˚(8.0m)

= 4.0 m

cosø =dx

d

dx = cosø(d) = cos30.0˚(8.0m)

= 6.9 m

Adjacent Component!

Opposite Component!

V

Θ

Vx

Vy

Adjacent component: Vx = VcosΘ

Opposite component: Vy = VsinΘ

tanΘ =Vy

Vx

Θ = tan-1 (Vy/Vx)

Be careful of the quandrant!

V = √ Vx2 + Vy

2

1) A man walks 5.0 km to the East and then walks 3.0 km to the North. What is his displacement from where he started?

2) What are the components of a vector displacement of 12.0 m @ 32.0˚?

3) If a student walks 56.0 m North and then turns West and walks another 85.0 m, what is his displacement?

4) Vector B has components of dx = -22 m and dy = - 33 m. What is the magnitude and direction of this vector? What is the magnitude and direction of – B ?

5) What is the resultant displacement when a box is moved 5.00 m in the x direction and then -7.50 m in the y direction?

6) What are the components of the vector shown below?

Aø

A = 27.3 m

Ø = 32.8˚

Adding Vectors Using Components

When adding two (or more) vectors, adding the components will give the components of

the Resultant vector:

A golfer on a flat green putts a ball 7.50 m in the Northeast direction, but misses the hole. He then putts the ball 2.30 m @ 38.0˚ South of straight East and sinks the putt for a bogey. What single putt would have saved par? d1 = 7.50 m @ 45.0˚

d2 = 2.30 m @ - 38.0˚

d1

45.0˚

-38.0˚d2

d1x = cos45.0˚(7.50 m) = 5.30 m

d1y = sin45.0˚(7.50 m) = 5.30 m

d2x = cos(-38.0˚)(2.30 m) = 1.81 m

d2y = sin(-38.0˚)(2.30 m) = -1.42 m

Head to Tail—On the head of the first goes the tail of the next vector!

dx = d1x + d2x = 5.30 m + 1.81 m = 7.11 m

dy = d1y + d2y = 5.30 m + (- 1.42 m) = 3.88 m

d = √ dx2 + dy

2 = √(7.11)2 + (3.88)2 = 8.10 m

Ø = tan-1 (dy / dx) = tan-1(3.88 / 7.11) = 28.6˚

d1

d2

d

ø

The single (resultant) putt:

d = 8.10 m @ 28.6˚

1) What is the resulting displacement when an object is moved 10.0 m to the North and then 5.0 m to the east?

2) A man leaves his house and walks 6.00 km to the West and then turns and walks 3.50 km to the South. What is his displacement?

3) A woman drives straight East for 65.0 km and then turns 30.0˚ North of East and drives another 33.0 km. What is her displacement?

4) A = 25.0 N @ 33.0˚ B = 57.7 N @ 152˚Find the resultant when vector A is added to vector B.

5) Add the following three vectors:

AB

Cø

αβ

A = 225 mα = 28.0˚

B = 275 mΒ = 56.0˚

C = 325 mø = 15.0˚

Relative Velocity

Velocities are vectors and add like vectors:

A plane flies through the air at a speed of 255 m/s. The air speed is 33.0 m/s. The velocity of the plane relative to the ground depends upon direction:

In each case, the plane is heading (pointed in that direction) South, but…

288 m/s 222 m/s 257 m/s @ 277˚

Remember: Default reference frame is Earth!

A boat travels at 12.0 m/s relative to the water and heads East across a river that flows North at 3.00 m/s. What is the speed and direction of the boat relative to the shore?

Vbw = 12.0 m/s @ 0˚ Vwg = 3.0 m/s @ 90.0˚

Vbw

Vwg

Vbg

ø

Vbg = (V12 + V2

2) = (12.02 + 3.002) = 12.4 m/s

Ø = tan-1(V2 / V1) = tan-1(3.00/12.0) = 14.0˚

Vbg = 12.4 m/s @ 14.0˚

1) A boat heads West across a stream that flows South. What is the velocity of the boat relative to the shore if it heads across with a speed of 8.3 m/s while the water flows South at 2.4 m/s?

3) A barge heading West down a still river travels at 5.0 m/s. A man walks across the barge from North to South at 2.0 m/s. What is the velocity of the man as viewed from a bridge above?

4) A boat wants to travel directly across a river that flows South at 3.0 m/s. If the boat travels at 7.0 m/s in still water, what heading must it take to go straight across? With what speed will the boat travel straight across?

5) An airplane has a velocity of 285 m/s @ 215˚ while flying through a crosswind. What is the heading of the plane? What is the velocity of the wind?

6) A man in a blue car traveling at 25.0 m/s @ 25.0˚ views a second red auto traveling at 32.0 m/s @ 215˚. What is the velocity of the red car relative the the man in the blue car?

7) An airplane flies at 225 m/s @ 45.0˚ North of east. A second plane flies at 175 m/s @ 35.0˚ South of North. What is the velocity of the the first plane relative to the second? What is the velocity of the second plane relative to the first?