## Pātai: Scalars vs Vectors

In pairs, think about and discuss the similarities and differences between these two questions:

- Mr Chu puts 40 apples inside a box, except Miss Nam eats two of them. What is the total number of apples inside the box?
- Mrs Carpenter lifts a plant off her desk with a force of $15N$ in the upwards direction, while the plant has a weight force of $5N$ acting down. What is the total force applied on the plant?

## What is a Vector?

**Scalar**= size only (e.g. mass)**Vector**= size + direction (e.g. velocity)

Discuss with your partner the difference between velocity and speed.

## Distance vs Displacement

**Distance**is the amount an object has moved- It is a scalar
- E.g. 3km

**Displacement**is the distance from start to finish in a straight line- It is a vector, because direction is also important
- E.g. 3km south west

### Pātai

Ella drives to Sumner beach in the weekend because it is far too hot. She drives $5km$ south and $10km$ west to get there.

- What is the total distance travelled by Ella?
- What is the total displacement of Ella?

#### Whakatika

- Distance: $d = 5km + 10km = 15km$
- Displacement: ?

## Akoranga 3 Te Whāinga Ako

- Introduce vector addition

Write the date and te whāinga ako in your book

### Scalar or Vector?

- Distance
- Displacement
- Speed
- Velocity
- Acceleration
- Momentum

- Energy
- Force
- Temperature
- Mass
- Work
- Power

When dealing with problems which involve vector quantities (e.g. calculating velocity, force, etc.), you must consider the size and direction.

Which means: **YOU MUST USE VECTOR CALCULATIONS and/or VECTOR DIAGRAMS.**

## Vectors

- Have both
**direction**and**magnitude** - Drawn as an arrow
- Drawn with a ruler
- Drawn to scale (on a grid, typically)

- Drawn head-to-tail
- Can be added an subtracted
- Use Pythagoras and SOH CAH TOA to find values

### Vector Addition

- To add vectors, we simply draw a the next vector from the
*arrowhead*of the previous one. - Draw the resultant vector from start to finish in a separate colour.
- Important: The resultant vector should be pointing from start position to finish position

### Vectors Worksheet

## Vector Addition Pātai Tahi

A car is driven 3 km east for 200 seconds, then 4 km south for 250 seconds, then 3 km west for 150 seconds.

- What is the total distance the car has travelled?
- What is the total displacement of the car?
- What is the average speed of the car?
- What is the average velocity of the car?

\begin{aligned} speed = \frac{distance}{time} \newline velocity = \frac{displacement}{time} \end{aligned}

## Akoranga 4 Mahi Tuatahi

## Ngā Whāinga Ako

- Complete practical vector addition examples
- Introduce vector subtraction
- Calculate vector $\Delta$

Write the date and ngā whāinga ako in your book

## Vector Addition Pātai Rua

- A bird flies $3km$ to the east and then $4km$ to the south. Find the resultant
**displacement**of the bird. - The bird takes $35min$ to complete the flight. Calculate its average
**speed**and**velocity**in meters per second.

### Vector Subtraction

Consider acceleration:

- Positive acceleration will increase speed
- Negative acceleration will decrease speed
- Pātai: What is different?
- Whakatika: The direction!

This works because of algebra:

\begin{aligned} & a - b = a + (-b) \newline & 1 - 4 = 1 + (-4) = -3 \end{aligned}

Vector subtraction is simply vector addition, where the subtracted vectors have their directions flipped.

## Vectors with $\Delta$

Velocity is a vector and a change ($\Delta$) is calculated like this:

\begin{aligned} & \Delta v = v_{f} - v_{i} \newline \end{aligned}

Pātai: Can we turn this into **vector addition**?

\begin{aligned} & \Delta v = v_{f} - v_{i} \newline & \Delta v = v_{f} + (-v_{i}) \newline \end{aligned}

#### Example / *Tauria Tahi*

A soccer ball collides with the crossbar of a goalpost at $5ms^{-1}$. It rebounds at $4ms^{-1}$ in the opposite direction away from the crossbar.

- Draw a vector diagram illustrating this
- Determine the ball’s change in velocity using the $\Delta v$ equation
- Remember to use
**K,U,F,S,S**

- Remember to use

## Ngā Whāinga Ako

- Practice Vector Addition and Subtraction
- Use trigonometry to give vector bearings in degrees

Write the date and ngā whāinga ako in your book

## Mahi Tuatahi

- Mr Le Sueur walked his dog up Mt Barossa on Sunday. He first walked 1.5km East (30min) and then 2.5km North (1hr 15min).
**Draw a vector diagram** - Calculate his
**average speed** - Calculate his
**average velocity** - Give the
**direction**he travelled using an angle and a cardinal direction (e.g. 10 deg north of west).

## Finding Directions

**SOH**: $sin(\theta) = \frac{opp}{hyp}$**CAH**: $cos(\theta) = \frac{adj}{hyp}$**TOA**: $tan(\theta) = \frac{opp}{adj}$

Make sure your calculator is in degrees NOT radians!

#### Textbook Questions

- ESA Study Guide: Page 108-109, Q3, Q4, Q5, Q7, Q13.
**Extra**: Homework booklet Q8

## Akoranga 6 Mahi Tuatahi

- Sarah passes the soccer ball to Kya at a speed of $15ms^{-1}$. Kya then passes it off to Atua at a speed of $5ms^{-1}$.
**Draw a vector diagram for this change in velocity**. - Calculate the change in velocity
- Calculate the angle of the change

## Akoranga 6 Mahi Tuatahi Rua

- A cyclist is heading down a hill at $14ms^{-1}$ S, before going around a sharp $90\degree$ bend, after which they are going $29ms^{-1}$ W. Draw the vectors $v_{f}$, $v_{i}$, $-v_{f}$ and $-v_{i}$
- Calculate their change in velocity (including direction) using a vector diagram, Pythagoreas theorem and trigonometry.

## Te Whāinga Ako

- Decompose vectors into horizontal and vertical components

Write the date and te whāinga ako in your book

## Vector Components

- Similarly to how we create a
*resultant vector*by adding two vectors, we can decompose a vector on an angle into its horizontal and vertical components. **Pātai**: Draw two vectors which*result*in $\vec{C}$ and form a**right-angled triangle**.

### Whakatika

- This is an important skill because the
**x**and**y**components of a vector are**independent**. - This means they do not influence each other, and often experience different accelerations

## Pātai

- Mr Le Sueur tosses a ball across the classroom with an initial velocity of $4ms^{-1}$ (hypotenuse), on an angle of $40\degree$.
**Draw a labelled vector diagram** - Write an equation to find the horizontal component of the velocity
- Write an equation to find the vertical component of the velocity

### Whakatika

\begin{aligned} v_{x} &= v \times cos(\theta) && \text{Horizontal} \newline v_{y} &= v \times sin(\theta) && \text{Vertical} \end{aligned}

## Textbook Pātai

- ESA Study Guide (2011): Page 108-109, Q8-12.
- ESA Study Guide (2008): Page 98-99, Q8-12.