Vectors

Physics

ObjectivesGraphical Method Vector AdditionVector AdditionRelative Velocity

Vectors and Scalars

A vector has magnitude as well as direction.

Some vector quantities: displacement, velocity, force, momentum

A scalar has only a magnitude.

Some scalar quantities: mass, time, temperature

1.3.1 Distinguish between vector and scalar quantities, and give examples of each. method. A vector is represented in print by a bold italicized symbol, for example, F.

1. The length of the line represents the magnitude and the arrow indicates the direction.2. The magnitude and direction of the vector is

clearly labeled.

The magnitude of a vector in a scaled vector diagram is depicted by the length of the arrow. The arrow is drawn precisely to length in accordance with a chosen scale.

Scaling!!!Scaling!!!

Direction!!!Direction!!!

Sometimes vectors will be directed due East or due North. However we will encounter vectors in all sorts of directions and be forced to find the angle!

Compass Coordinate System

Compass Coordinate System

• Δx = 30 m @ 20º E of N

• V = 20 m/s @ 30º W of N

• a = 10 m/s2@ 40º W of S

• F = 50 N @ 10º S of E

Navigational System?

Use the same scale for all vector magnitudes

E

S

N

W

To Draw direction:

Ex. 20º E of N: Start w/ North and go 20° East

All these planes have the same reading on their speedometer. (plane speed not speed with respect to the ground (actual speed)

What factor is affecting their velocity?

A B C

Easy Adding…

The resultant is the vector sum of two or more vectors.

1. Select an appropriate scale (e.g., 1 cm = 5 km)

2. Draw and label 1st vector to scale.

*The tail of each consecutive vector begins at the head of the most recent vector*

3. Draw and label 2nd vector to scale starting at the head of the 1st vector.

4. Draw the resultant vector (the summative result of the addition of the given vectors) by connecting the tail of the 1st vector to the head of the 2nd vector. (initial to final pt.)

5. Determine the magnitude and direction of the resultant vector by using a protractor, ruler, and the indicated scale; then label the resultant vector.

A. Tailwind

(with the wind)

120 km/h20 km/h

80 km/h

100 km/h

100 km/h

20km/h

=

B. Headwind

(against the wind)

=

80 km/h

100 km/h

60 km/h

Resultant

C. 90º crosswindUsing a ruler and your scale, you can determine the magnitude of the resultant vector. Or you could use the Pythagorean Theorem.

Then using a protractor, you can measure the direction of the resultant vector. Or you could use trigonometry to solve for the angle.

0o

N

180o

270o

An airplane is flying 200mph at 50o N of E. Wind velocityis 50 mph due S. What is the velocity of the plane?

Scale: 50 mph = 1 inch

E

N

W

S

An airplane is flying 200mph at 50o N of E. Wind velocityis 50 mph due S. What is the velocity of the plane?

Scale: 50 mph = 1 inch

E

N

W

S

An airplane is flying 200mph at 50o N of E. Wind velocityis 50 mph due S. What is the velocity of the plane?

Scale: 50 mph = 1 inch

200 mph

50 mph

VR = 165 mph @ 40° N of E

2. Find the resultant velocity vector of the two velocity vectors below.

700 m/s @35 degrees E of N; 1000 m/s @ 30 degrees N of W

V1

Vr

V2

Relative VelocityWe already considered relative speed in one dimension; it is similar in two dimensions except that we must add and subtract velocities as vectors.

Each velocity is labeled first with the object, and second with the reference frame in which it has this velocity. Therefore, vWS is the velocity of the water in the shore frame, vBS is the velocity of the boat in the shore frame, and vBW is the velocity of the boat in the water frame.

Relative VelocityIn this case, the relationship between the three velocities is:

(3-6)

Relative Velocity

Relative Velocity-Sample Problem

• A boat is capable of traveling at 5 m/s in still water. The boat attempts to cross a river which runs at 3 m/s downstream. What is the boats velocity as it crosses the river?

5 m/s

3 m/sR

Relative Velocity-Sample Problem

s

mR

s

mR

s

m

s

mR

s

m

s

mR

83.5

34

925

)3()5(

2

22

2

2

22

222

o

smsmsmsm

96.30

)5

3(tan

5

3tan

1

Answer: 5.83 m/s at 30.96o downstream

Relative Velocity-Sample Problem

• If the river is 1 km wide, how long will it take the boat to cross the river?

This problem turns into a simple linear motion problem. The boat has a velocity across the river of 5 m/s and the distance straight across the river is 1 km.

Relative Velocity-Sample Problem

smm

t

tkmm

km

s

m

t

dv

5

1000

11000

15

Answer: 200 seconds

Relative Velocity-Sample Problem

• How far downstream does the boat end up?

This also turns into a simple linear motion problem. Your boat is moving downstream at 3 m/s and it is in the water for 200 seconds.

Relative Velocity-Sample Problem

sec2003

sec2003

s

md

d

s

mt

dv

Answer: 600 m

Relative Velocity-Sample Problem

• A railway car is moving at 2 m/s and a man starts to walk from the back of the car to the front at 1 m/s. What is his velocity relative to the ground?

Answer: 3 m/s

An airplane flies at an engine speed of 100 m/s at 50º W of S into a wind of 30 m/s at 200 E of N. What is the airplane’s resultant velocity?

Solve using the components method!!

How far has the plane traveled after 1 hr??

Answer: 75.52 m/s @ 28.54˚ S of W

168.89 miles per 1 hour

You Try!!!You Try!!!

A motor boat traveling 4.0 m/s, East encounters a current traveling 3.0 m/s, North.

a. What is the resultant velocity of the motor boat?

b. If the width of the river is 80 meters wide, then how much time does it take the boat to travel shore to shore?

c. What distance downstream does the boat reach the opposite shore?