Today, we will have a short review on vectors and projectiles and then have a quiz.

You will need a calculator, a clicker and some scratch paper for the quiz.

VectorsVector: a quantity that has both magnitude (size) and direction

Examples: displacement, velocity, acceleration

Scalar: a quantity that has no direction associated with it, only a magnitudeExamples: distance, speed, time, mass

Vectors are represented by arrows.

The length of the arrow represents the magnitude (size) of the vector.

And, the arrow points in the appropriate direction.

20 m/s 50 m/s

NW

East

Adding vectors graphically

1. Without changing the length or the direction of any vector, slide the tail of the second vector to the tip of the first vector.

2. Draw another vector, called the RESULTANT, which begins at the tail of the first vector and points to the tip of the last vector.

+

Subtracting vectors graphically

1. First, reverse the direction of the vector you are subtracting. Then, without changing the length or the direction of any vector, slide the tail of the second vector to the tip of the first vector.

2. Draw another vector, called the RESULTANT, which begins at the tail of the first vector and points to the tip of the last vector.

Adding co-linear vectors(along the same line)

A = 8 m B = 4 m

A + B = R = 12 m

C = 10 m/s D = - 3 m/s

C + D = 10 + (-3) = R = 7 m/s

Airplane Tailwind

Airplane Headwind

Adding perpendicular vectors

How could you find out the length of the RESULTANT?

Since the vectors form a right triangle, use the PYTHAGOREAN THEOREM

A2 + B2 = C2

10 m

6 m 67.11610C 22 11.67 m

A boat capable of going 3 m/s in still water is crossing a river with a current of 5 m/s. If the boat points straight across the river, where will

it end up- straight across the river or downstream?

Downstream, because that is where the current will carry it even as it goes across!

What is the resultant velocity?

Resultant velocity

Sketch the 2 velocity vectors:the boat’s velocity, 5 m/s and the water’s velocity, 3 m/s

Resultant velocity,

sm8.553v 22

Vb = 3 m/s

Vw = 5 m/s

Resultantvelocity

A long walk…A hiker walked 3 km East, then 8 km North, then 5 km West, then 4 km South. What was her displacement from her starting position?

Look at East/West motion. Then look at North/South motion.

3 km East + 5 km West = 2 km West8 km North + 4 km South = 4 km North

Now add these two perpendicular vectors together:2 km West + 4 km North =

= 4.47 km Northwest

Vector COMPONENTSEach vector can be described to

terms of its x and y components.

X (horizontal) component

Y (vertical)component

If you know the lengths of the x and y components, you can calculate the length of the vector using the Pythagorean.

Drawing the x and y components of a vector is called “resolving a vector into its components”

Make a coordinate system and slide the tail of the vector to the origin.

Draw a line from the arrow tip to the x-axis.

The components may be negative or positive or zero.

X component

Y component

Calculating the componentsHow to find the length of the components if you know the magnitude and direction of the vector.

Sin q = opp / hypCos q adj / hypTan q = opp / adj

SOHCAHTOA

A

Ax

Ay

q

= A sin q

= A cos q

= 12 m/s

= 35 degrees= 12 cos 35 = 9.83 m/s

= 12 sin 35 = 6.88 m/s

Are these components positive or negative?

V = 22 m/s

q = 50˚What is vx?Vx = - v cos q˚Vx = -22 cos 50˚Vx = - 14.14 m/s What is vy?Vy = v sin q˚Vy = 22 sin 50˚Vy = 16.85 m/s

Finding the angleSuppose a displacement vector has an x-component of 5 m and a y-component of - 8 m. What angle does this vector make with the x-axis?

q = ?We are given the side adjacent to the angle and the side opposite the angle.Which trig function could be used?Tangent q = opposite ÷ adjacent

Use “arc tangent” to determine the angle “tan -1 ”

Therefore the angle q = tan -1 (8 ÷ 5)q = 57.99 degrees below the positive x-axis

Adding Vectors by components

Slide each vector to the origin.Resolve each vector into its x and

y componentsThe sum of all x components is

the x component of the RESULTANT.

The sum of all y components is the y component of the RESULTANT.

Using the components, draw the RESULTANT.

Use Pythagorean to find the magnitude of the RESULTANT.

Use inverse tan to determine the angle with the x-axis.

qb

A

B

R

x y

A = 18, q 20 degreesB = 15, b = 40 degrees

18 cos 20 18 sin 20

-15 cos 40 15 sin 40

5.42 15.8

7.168.1542.5R 22

a tan-1(15.8 / 5.42) = 71.1 degrees above the positive x-axis

aA

B

Walking on a train- relative velocity

What is the velocity of the man on the train relative to the person standing outside on the ground?

A boat capable of going 5 m/s in still water is crossing a river with a current of 3 m/s. If the boat points straight across the river, where will

it end up- straight across the river or downstream?

Downstream, because that is where the current will carry it even as it goes across!

What is the resultant velocity?

Resultant velocity

Sketch the 2 velocity vectors:the boat’s velocity, 5 m/s and the river’s velocity, 3 m/s

Resultant velocity,

sm8.553v 22

5 m/s

3 m/s

Resultantvelocity