Physics Lesson 5

Two Dimensional Motion and Vectors

Eleanor Roosevelt High School

Mr. Chin-Sung Lin

Two Dimensional Motion and Vectors

Scalars & Vectors

Vector Representation

One-Dimensional Vector Addition

Two-Dimensional Vector Addition

Vector Resolution

Vector Addition Through Resolution

Vector Application: Relative Velocity

Scalars & Vectors

Scalars & Vectors

Comparison of Scalars & Vectors

Scalars Vectors

Magnitude Magnitude

Direction

Physical Quantities

Comparison of Scalars & Vectors

Scalars Vectors

Magnitude Magnitude

Direction

Physical Quantities

3 m/s

60o

Scalars & Vectors

Examples of Scalars & Vectors

Scalars Vectors

Physical Quantities

Scalars & Vectors

displacement

velocity

acceleration

force

distance

speed

acceleration

mass

Vector Representation

Arrows

An arrow is used to represent the magnitude and direction of a vector quantity

Magnitude: the length of the arrow

Direction: the direction of the arrow

Head

Tail

Magnitude

Direction

Vector Representation

Equality of Vectors

Vectors are equal when they have the same magnitude and direction, irrespective of their point of origin

Magnitude

Direction

Vector Representation

Negative Vectors

A vector having the same magnitude but opposite direction to a vector

A

Vector Representation

- A

One-Dimensional Vector Addition

Vector Addition (Same Direction)

The result of adding two vectors (resultant) with the same direction is the sum of the two magnitudes and the same direction

One-Dimensional Vector Addition

10 m

5 m 5 m

Vector Addition (Opposite Directions)

The result of adding two vectors (resultant) with opposite directions is the difference of the two magnitudes and the direction of the longer one

One-Dimensional Vector Addition

10 m

-5 m

5 m

Two-Dimensional Vector Addition

Vector Addition (Parallelogram Method)

The resultant is the diagonal of the parallelogram described by the two vectors

Two-Dimensional Vector Addition

ResultantB

A

Vector Addition (Head-Tail Method)

Many vectors can be added together by drawing the successive vectors in a head-to-tail fashion. The resultant is from the tail of the first vector to the head of the last vector

Two-Dimensional Vector Addition

ResultantB

A

Vector Subtraction

One vector subtracts another vector is the same as one vector adds another negative vector

Two-Dimensional Vector Addition

A

A – B = A + (-B)

B

Vector Subtraction

One vector subtracts another vector is the same as one vector adds another negative vector

Two-Dimensional Vector Addition

Resultant- B

A

A – B = A + (-B)

Vector Resolution

Component Vectors

Any vector can be resolved into two component vectors (vertical and horizontal components) at right angle to each other

Vector Resolution

Horizontal component

Vector Vertical component

Component Vectors

The process of determining the components of a vector is called vector resolution

Vector Resolution

Horizontal component

Vector Vertical component

Calculate Component Vectors

The magnitude of the horizontal component vx = v cos θ

The magnitude of the vertical component vy = v sin θ

Vector Resolution

Vx = V cos θ

VVy = V sin θ

θ

Two-dimensional vector addition

through vector resolution

Two-Dimensional Vectors Addition

Resolve vectors into horizontal and vertical components

Add all the horizontal components of the vectors

Add all the vertical components of the vectors.

Find the final resultant by adding the horizontal and vertical components of the final resultant

Vector Addition through Resolution

Two-Dimensional Vectors Addition

Vector Addition through Resolution

Ax

AAy

By

R

Bx

B

Rx

Ry

Two-Dimensional Vectors Addition

Vector Addition through Resolution

34.6 m/s

40.0 m/s20.0 m/s

-26.0 m/s

-15.0 m/s

30.0

m/s

19.6 m/s

-6.0 m/s

30o

60o

20.5 m/s

-16.9o

34.6 m/s – 15.0 m/s = 19.6 m/s

20.0 m/s – 26.0 m/s = -6.0 m/s

tan-1 (-6.0 m/s /19.6 m/s) = -16.9o

sqrt (19.62 + 6.02) m/s = 20.5 m/s-30.0 m/s sin (60o) = -26.0 m/s

-30.0 m/s cos (60o) = -15.0 m/s

40.0 m/s sin (30o) = 20.0 m/s

40.0 m/s cos (30o) = 34.6 m/s

Vector Application:Relative Motion

Relative Velocity

Relative velocity is the vector difference between the velocities of two objects in the same coordinate system

Vector Application

Relative Velocity

For example, if the velocities of particles A and B are vA and vB respectively in the same coordinate system, then the relative velocity of A with respect to B (also called the velocity of A relative to B) is vA – vB

VA VA – VB

VB

Vector Application

Relative Velocity

The relative velocity vector calculation for both one- and two-dimensional motion are similar

The velocity vector subtraction (vA – vB ) can be viewed as vector addition (vA + (–vB))

Vector Application

VA VA +(–VB)

-VB

Relative Velocity

Conversely the velocity of B relative to A is vB – vA

Vector Application

VA VB – VA

VB

Q & A

The End