Vectors

Pearland ISD Physics

Scalars and Vectors

• A scalar quantity is one that can be described by a single number:– Examples:

• temperature, speed, mass

• A vector quantity deals inherently with both magnitude and direction:– Examples:

• Acceleration, velocity, force, displacement

Identify the following as scalars or vectors – not in your notes

• The acceleration of an airplane as it takes off

• The number of passengers and crew on the airplane

• The duration of the flight• The displacement of the flight• The amount of fuel required for the

flight scalar

vector

scalarscalar

vector

Representing vectors

• Vectors can be represented graphically.– The direction of the arrow is the direction of

the vector. – The length of the arrow tells the magnitude

• Vectors can be moved parallel to themselves and still be the same vector.– Vectors only tell magnitude (amount) and

direction, so a vector can starts anywhere

4 N 8 N

Adding vectors

• The sum of two vectors is called the resultant.

• To add vectors graphically, draw each vector to scale.

• Place the tail of the second vector at the tip of the first vector. (tip-to-tail method)

• Vectors can be added in any order.• To subtract a vector, add its opposite.

Vector Addition and Subtraction

• Often it is necessary to add one vector to another.

Vector Addition and Subtraction

5 m 3 m

8 m

Adding Vector Problem

A parachutist jumps from a plane. He has not pulled is parachute yet. His weight or force is 800 N downward. The wind is applying a small drag force of 50 N upward. What is the vector sum of the forces acting on him?

750 N downward

Adding perpendicular vectors

• Perpendicular vectors can be easily added and we use the Pythagorean theorem to find the magnitude of the resultant.

• Use the tangent function to find the direction of the resultant.

Vector Addition and Subtraction

1.6 Vector Addition and Subtraction

2.00 m

6.00 m

Vector Addition and Subtraction

2.00 m

6.00 m

222 m 00.6m 00.2 R

R

m32.6m 00.6m 00.2 22 R

4.180.6

0.2tan 1

Adding perpendicular Vectors – this slide is not in your notes

Note: in this example, In order to use the tip-to-tail Method, Vector B must be moved.

Vector Addition and Subtraction This slide is not in your notes

When a vector is multiplied by -1, the magnitude of the vector remains the same, but the direction of the vector is reversed.

YOU MAY WANT TO MAKE A NOTE OF THIS!!

Multiplying Vectors

• A vector multiplied or divided by a scalar results in a new vector.– Multiplying by a positive number changes

the magnitude of the vector but not the direction.

– Multiplying by a negative number changes the magnitude and reverses the direction.

Resolving vectors into components.• Any vector can be resolved, that is,

broken up, into two vectors, one that lies on the x-axis and one on the y-axis.

1.7 The Components of a Vector

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component vector thecalled are and

r

yx

y

x

1.7 The Components of a Vector

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AA

A

yx

that soy vectoriall together add and

axes, and the toparallel are that and vectors

larperpendicu twoare of components vector The

yxyx

Calculate the following: Practice!!

• A roller coaster moves 85 meters horizontally, then travels 45 meters at an angle of 30.0° above the horizontal. What is its displacement from its starting point?

• A pilot sets a plane’s controls, thinking the plane will fly at 2.50 X 102 km/hr to the north. If the wind blows at 75 km/hr toward the southeast, what is the plane’s resultant velocity?

RELATIVE VELOCITYSECTION: 4

Relative Motion

• The motion of an object can be expressed from different points of view

• These points of view are known as “frames of reference”

• Depending on the frame of reference used, the description of the motion of the object may change

Relative Velocity (motion of objects is independent of each other)

• Velocity of A relative to B: using the ground as a reference frame

• The LanguagevAG : v of A with respect to G (ground) = 40 m/svBG : v of B with respect to G = 30 m/svAB : v of A with respect to B:

BGAGAB vvv

smsmAB /30/40 v

smAB /10v

Example 1

• The white speed boat has a velocity of 30km/h,N, and the yellow boat a velocity of 25km/h, N, both with respect to the ground. What is the relative velocity of the white boat with respect to the yellow boat?

YGWGWY vvv

hkmhkmWY /25/30 v

hkmWY /5v

Example 2- The Bus Ride Lets do this together

A passenger is seated on a bus that is

traveling with a velocity of 5 m/s, North.

If the passenger remains in her seat,

what is her velocity:

a) with respect to the ground?

b) with respect to the bus?

Example 2 –continuedLets do this together

The passenger decides to approach the driver with a velocity of 1 m/s, N, with respect to the bus, while the bus is moving at 5m/s, N.

What is the velocity of the passenger with respect to the ground?

Resultant Velocity (motion of objects is dependent on each other)

The resultant velocity is the net velocity

of an object with respect to a reference

frame.

Example 3- Airplane and Wind - relative to the groundAn airplane has a velocity of 40 m/s, N, in still air. It is facing a headwind of 5m/s with respect to the ground.

What is the resultant velocity of the airplane?

WGPGPG vvv

smsmPG /5/40 v

smPG /35v

Frame of reference

• What is this guy’s velocity?

• What about now?

Frame of reference

• What about now?

Relative Velocity:Example 4: Crossing a River

The engine of a boat drives it across a river that is 1800m wide.The velocity of the boat relative to the water is 4.0m/s directed perpendicular to the current. The velocity of the water relativeto the shore is 2.0m/s.

A)What is the velocity of the boat relative to the shore?

B) How long does it take for the boat to cross the river?

Relative Velocity

• G: • U: • E: (Pythagorean Theorem)

• S:

• S:

2222 sm0.2sm0.4 WSBWBS vvv

WSBWBS vvv

63/0.2

/0.4tan 1

sm

sm

sm /0.4BWv sm /0.2WSv

?BSv

sm5.4

Relative Velocity

• G: • U:

• E: (Kinematics)

• S:

• S: s 450sm4.0

m 1800t

?t

t

x

v

v

xt