Scalars vs Vectors

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?
• Ms 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 and direction (e.g. velocity)

Discuss with your partner the difference between velocity and speed.

Example: 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 nice and hot outside. 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: ?

Scalar or Vector?

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 DIAGRAMS (trigonometry) when working in two dimensions!

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

Why can’t we add vectors algebraically?

Vector Addition Example

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

1. What is the total distance the car has travelled?
2. What is the total displacement of the car?
3. What is the average speed of the car?
4. What is the average velocity of the car?

\[ \begin{aligned} speed = \frac{distance}{time} \cr velocity = \frac{displacement}{time} \end{aligned} \]

Vectors Worksheet

Mahi Tuatahi

1. https://quizlet.com/au/566254686/vectors-and-scalars-flash-cards/

Ngā Whāinga Ako

1. Complete practical vector addition examples
2. Introduce vector subtraction
3. Calculate vector \(\Delta\)

Write the date and ngā whāinga ako in your book

Vector Subtraction

Consider acceleration:

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

Summary: Vector subtraction is simply vector addition, where the subtracted vectors have their directions inverted.

Vectors with \(\Delta\)

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

\[ \begin{aligned} \Delta v &= v_{f} - v_{i} \cr \end{aligned} \]

Pātai: Looks like vector subraction to me. Can we turn it into addition?

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

Example / Tauria

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.

1. Draw a vector diagram illustrating this
2. Determine the ball’s change in velocity using the \(\Delta v\) equation
   • Remember to use K,U,F,S,S

Practice / Whakawai

• ESA Activity 9A Q11-14
• Course Manual handout page 3-4

Textbook Questions

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

Answer / Whakatika

\[ \begin{aligned} \Delta v^{2} &= v_{f}^{2} + v_{i}^{2} \cr \Delta v &= \sqrt{v_{f}^{2} + v_{i}^{2}} \cr \Delta v &= \sqrt{15^{2} + 5^{2}} \cr \Delta v &= \sqrt{250} = 15.81ms^{-1} \cr\cr tan(\theta) &= \frac{opp}{adj} \cr \theta &= tan^{-1}(\frac{15}{5}) = 75.6\degree \end{aligned} \]

Te Whāinga Ako

1. Decompose vectors into horizontal and vertical components

Write the date and te whāinga ako in your book

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 (e.g. in projectile motion)

Whakatika

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

Textbook Pātai

• ESA Study Guide (2011): Page 108-109, Q1-14.
• ESA Study Guide (2008): Page 98-99, Q1-14.