Skip to main content

Vectors

Pātai: Scalars vs Vectors

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



What is a Vector?

Discuss with your partner the difference between velocity and speed.


Distance vs Displacement


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.


Whakatika

Source


Akoranga 3 Te Whāinga Ako

  1. Introduce vector addition

Write the date and te whāinga ako in your book


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 CALCULATIONS and/or VECTOR DIAGRAMS.


Vectors



Vector Addition

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:


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

Make sure your calculator is in degrees NOT radians!


Textbook Questions


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


Whakatika


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