Vectors

12PHYS - Mechanics

Finn LeSueur

2021

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: ?
Source

Akoranga 3 Te Whāinga Ako

  1. 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
Vector Addition

Vectors Worksheet

Source

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.

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

Akoranga 4 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 Addition Pātai Rua

  1. A bird flies \(3km\) to the east and then \(4km\) to the south. Find the resultant displacement of the bird.
  2. 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!
Vector Subtraction

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.

  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

Ngā Whāinga Ako

  1. Practice Vector Addition and Subtraction
  2. Use trigonometry to give vector bearings in degrees

Write the date and ngā whāinga ako in your book

Mahi Tuatahi

  1. 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
  2. Calculate his average speed
  3. Calculate his average velocity
  4. 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

  1. 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.
  2. Calculate the change in velocity
  3. Calculate the angle of the change

Akoranga 6 Mahi Tuatahi Rua

  1. 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 \(-v_{i}\)
  2. Calculate their change in velocity (including direction) using a vector diagram, Pythagoreas theorem and trigonometry.

Te Whāinga Ako

  1. 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
Vector Decomposition

Pātai

Vector Decomposition
  1. 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
  2. Write an equation to find the horizontal component of the velocity
  3. Write an equation to find the vertical component of the velocity

Whakatika

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

Textbook Pātai

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