VECTORS AND TWO-DIMENSIONAL MOTION

Properties of Vectors

Vectors vs. Scalars

• Physical quantities can be categorized as:

Scalar quantities or Vector quantities

Magnitude (with appropriate units)

Magnitude (with appropriate units)

Direction+

Examples:TemperatureMassTime

Velocity

Examples:

DisplacementAcceleration

Adding Vectors graphically

• If we can add scalars then we can add vectors

They must have the same units

m m+

m m/s+

m m

They must be connected from tip to tail Resultant (R)= A + B

TRIANGLE METHOD OF ADDITION

A

B

A

B

R

Commutative Law of Addition

AB

R

SUBTRACTING VECTORS

The negative of a Vector

B -B

A-B

A-B

R

ADDITION VS. SUBTRACTION

ADDITION A B+

R

A - B

R =0

A

BR A

-B

R

SUBTRACTION

EXAMPLE PROBLEM (We solve)

• GOAL: Find the sum of two vectors

A car travels 20.0 km due north and then 35.0 km in a direction 60° west of north. Find the magnitude and direction of the resultant vector. This vector is called the car’s resultant displacement (measure).

Vector A has a magnitude of 29 units and points in the positive y-direction. When vector B is added to A, the resultant vector A + B points in the negative y-direction with a magnitude of 14 units. Find the magnitude and direction of B.

EXAMPLE PROBLEM (We solve)

COMPONENTS OF A VECTOR

A

AX

AY

θ

EXAMPLE PROBLEM• Goal: Find vector components, given a magnitude

and direction

Find the horizontal and vertical components of the 1.00x10^2 m displacement of a superhero who flies from the top of a tall building along the path shown.

30.0°

100m

PRACTICE PROBLEM

• Goal: Find the resultant vector, given its components

Suppose instead the superhero leaps in the other direction along a displacement vector B to the top of a flagpole where the displacement components are given by Bx= -25.0m and By = 10.0m. Find the magnitude and direction of the displacement vector.

Practice Problem

Suppose the superhero had flown 150m at a 120° angle with respect to the positive x-axis. Find the components of the displacement vector.

Practice Problem

Suppose instead the superhero had leaped with a displacement having an x-component of 32.5 m and a y-component of 24.3 m. Find the magnitude and direction of the displacement vector.

x-components with x-components onlyRx = Ax + Bx

ADDING VECTORS ALGEBRAICALLY

y-components with y-components onlyRy = Ay + By

Subtracting vectors is the same thing, because subtracting is the same as adding the negative of one vector to the other.

PRACTICE PROBLEM

• Goal: Add vectors algebraically and find the resultant vector

A hiker begins a trip by first walking 25.0 m 45.0° sourth of east from her base camp. On the second day she walks 40.0 km in a direction 60.0° north of east, at which point she discovers a forest ranger’s tower.

a) Determine the components of the hiker’s displacements in the first and

second days.

b) Determine the components of the hiker’s total displacement for the trip

c) Find the magnitude and direction of the displacement from base camp

A

AY

B

R

45.0°60.0°

PRACTICE PROBLEM

A cruise ship leaving port travels 50.0 km 45.0° north of west and then 70.0 km at a heading 30.0° north of east. Find:

a) The ship’s displacement vector andb) The displacement vector’s magnitude and

direction.

EXAMPLE PROBLEM (You solve)

Vector A has a magnitude of 8.00 units and makes an angle of 45.0° with the positive x-axis. Vector B also has a magnitude of 8.00 units and is directed along the negative x-axis. Using graphical methods, find

(a) the vector sum A + B.(b) The vector difference A - B

EXAMPLE PROBLEM (You+Partner solve)

Vector A is 3.00 units in length and points along the positive x-axis. Vector B is 4.00 units in length and points along the negative y-axis. Use graphical methods to find the magnitude and direction of the vectors

a) A + Bb) A - B

HOMEWORK

3 easy problems + one challenging problem Posted on website.

Homework problem 1

Each of the displacement vectos A and B shown has a magnitude of 3.00m. Find

a) A + Bb) A – Bc) B – Ad) A – 2B

30°

3.00m3.00m

0

B

B

Homework Problem 2

A roller coaster moves 200 ft horizontally and then rises 135 ft at an angle of 30.0° above the horizontal. Next, it travels 135 ft at an angle of 40.0° below the horizontal. Find the roller coaster’s displacement from its starting point to the end of this movement.

A plane flies from base camp to lake A, a distance of 280 km at a direction of 20.0° north of east. After dropping off supplies, the plane flies to lake B, which is 190 km and 30.0° west of north from lake A. Graphically determine the distance and direction from lake B to the base camp.

Homework Problem 3

Homework Problem 4

An airplane flies 200 km due west from city A to city B and then 300 km in the direction of 30.0° north of west from city B to city C.

a) In straight-line distance, how far is city C from city A?

b) Relative to city A, in what direction is city C?

Homework Problem 5

A jogger runs 100 m due west, then changes direction for the second leg of the run. At the end of the run, she is 175 m away from the starting point at an angle of 15.0° north of west. What were the direction and length of her second displacement?

Homework Problem 6

A man lost in a maze makes three consecutive displacements so that at the end of his travel he is right back where he started. The first displacement is 8.00 m westward and the second is 13.0 m northward. What are the magnitude and direction of the third displacement?