Welcome to A.M. Online!

1 – Introduction to Vectors

2 – 1D Horizontal Motion

3 – Motion due to Gravity

4 – Continuous Change

5 – Forces and Systems

6 – Introduction to Energy and Momentum

7 – Impacts and Collisions

8 – Uniform Circular Motion

9 – Difference Equations

10 – Introduction to Graph Theory

11 – Path Optimisation

12 – MST Optimisation

13 – Project Optimisation

Mathematical Modelling Project

Revision Material

HL Sample Papers

HL Exam Questions by Topic

OL Sample Papers

OL Exam Questions by Topic

Weekly Grinds

Exercise Set 1A Solutions

Exercise Set 1A

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This set of questions will primarily test a student’s knowledge of the following lessons:

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1.1 – Scalar and Vectors

1.2 – Visual Representation of Vectors

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We recommend that students of all levels attempt the set below by following the suggested approach as stated within that set.

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For each question, we recommend that each student uses the following approach until they feel confident that they fully understand the answer:

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Answer \(\rightarrow\) Solution \(\rightarrow\) Walkthrough \(\rightarrow\) Teacher Chat

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Exercise Set 1A

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These questions are considered beginner to intermediate Ordinary Level questions.

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We recommend that students move on to the next lesson only once they have answered \(\mathbf{6}\) questions fully correct within this set.

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If a student reaches the end of this set without getting \(6\) questions fully correct, we recommend that they first quickly review the lessons stated above before moving on to the next lesson.

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If a student still feels doubtful that they are fully prepared for these questions if they were to appear on their exam, we suggest that they book a grind with Mr. Kenny.

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Question 1

Consider the following vectors.

Which of these vectors

(a) are directed west?

(b) are directed east?

(c) has the largest magnitude?

(d) has the smallest magnitude?

(a) \(\vec{c}\) only

(b) \(\vec{a}\) and \(\vec{b}\)

(c) \(\vec{c}\)

(d) \(\vec{b}\)

(a) \(\vec{c}\) only

(b) \(\vec{a}\) and \(\vec{b}\)

(c) \(\vec{c}\)

(d) \(\vec{b}\)

(a) As the arrow representing \(\vec{c}\) is pointing to the left, that vector is directed west.

(b) As the arrows representing \(\vec{a}\) and \(\vec{b}\) are pointing to the right, both of these vectors are directed east.

(c) As the arrow representing  \(\vec{c}\) is the longest, \(\vec{c}\) has the largest magnitude of the three.

(d) As the arrow representing  \(\vec{b}\) is the shortest, \(\vec{b}\) has the smallest magnitude of the three.

Video Walkthroughs are currently in the process of being gradually uploaded, with that process expected to be finished by the end of October.

In the meantime, if you don't understand something in the Walkthrough above, feel free to reach out to Mr. Kenny on Teacher Chat!

Question 2

Vector \(\vec{a}\) has a magnitude of \(12\) metres and a direction east.

State the magnitude of this vector:

(a) in centimetres

(b) in millimetres

(c) in kilometres

(a) \(1{,}200\mbox{ cm}\)

(b) \(12{,}000\mbox{ mm}\)

(c) \(0.012\mbox{ km}\)

(a)

\begin{align}12\times 100=1{,}200\mbox{ cm}\end{align}

(b)

\begin{align}12\times 1{,}000=12{,}000\mbox{ mm}\end{align}

(c)

\begin{align}12\times \frac{1}{1{,}000}=0.012\mbox{ km}\end{align}

(a) As there are \(100\) centimetres in a metre, there are \(12\times 100=1{,}200\) centimetres in \(12\) metres.

(b) As there are \(1{,}000\) millimetres in a metre, there are \(12\times 1{,}000=12{,}000\) millimetres in \(12\) metres.

(c) As there are \(1{,}000\) metres in a kilometre, there are \(\dfrac{1}{1{,}000}=0.001\) kilometres in a metre.

Therefore, there are \(12\times 0.001=0.012\) kilometres in \(12\) metres.

Video Walkthroughs are currently in the process of being gradually uploaded, with that process expected to be finished by the end of October.

In the meantime, if you don't understand something in the Walkthrough above, feel free to reach out to Mr. Kenny on Teacher Chat!

Question 3

Consider the following vector \(\vec{a}\).

Using a ruler:

(a) Redraw \(\vec{a}\).

(b) Construct a vector \(\vec{b}\) whose magnitude is half that of \(\vec{a}\) and that points in the same direction as \(\vec{a}\).

(c) Construct a vector \(\vec{c}\) whose magnitude is the same as \(\vec{a}\) and that points in the opposite direction to \(\vec{a}\).

(a)

(b)

(c)

(a)

(b)

(c)

(a) This vector has been redrawn using a ruler as below.

(b) As the magnitude of \(\vec{b}\) is half that of \(\vec{a}\), the arrow representing \(\vec{b}\) will be half as long as the arrow representing \(\vec{a}\).

As \(\vec{b}\) points in the same direction as \(\vec{a}\), the arrow representing \(\vec{b}\) points in the same direction as the arrow representing \(\vec{a}\).

(c) As the magnitude of \(\vec{c}\) is the same as \(\vec{a}\), the arrow representing \(\vec{c}\) will be the same length as the arrow representing \(\vec{a}\).

As \(\vec{c}\) points in the opposite direction to \(\vec{a}\), the arrow representing \(\vec{b}\) points in the opposite direction to the arrow representing \(\vec{a}\).

Video Walkthroughs are currently in the process of being gradually uploaded, with that process expected to be finished by the end of October.

In the meantime, if you don't understand something in the Walkthrough above, feel free to reach out to Mr. Kenny on Teacher Chat!

Question 4

The speed limit of most Irish motorways is \(120\) kilometres per hour.

Using a calculator, state this speed limit in metres per second.

\(33\dfrac{1}{3}\mbox{ m/s}\)

\begin{align}120\mbox{ km/h}&= \frac{120 \mbox{ kilometres}}{1\mbox{ hour}}\\&=\frac{120\times 1{,}000\mbox{ metres}}{1\times 3{,}600\mbox { seconds}}\\&=\frac{120{,}000}{3{,}600}\frac{\mbox{metres}}{\mbox{second}}\\&=33\frac{1}{3}\mbox{ m/s}\end{align}

There are \(1{,}000\) metres in a kilometre and there are \(60\times 60=3{,}600\) seconds in an hour.

Therefore, we can rewrite this speed limit as follows:

\begin{align}120\mbox{ km/h}&= \frac{120 \mbox{ kilometres}}{1\mbox{ hour}}\\&=\frac{120\times 1{,}000\mbox{ metres}}{1\times 3{,}600\mbox { seconds}}\\&=\frac{120{,}000}{3{,}600}\frac{\mbox{metres}}{\mbox{second}}\\&=33\frac{1}{3}\mbox{ m/s}\end{align}

where the final answer was found using a calculator.

Video Walkthroughs are currently in the process of being gradually uploaded, with that process expected to be finished by the end of October.

In the meantime, if you don't understand something in the Walkthrough above, feel free to reach out to Mr. Kenny on Teacher Chat!

Question 5

(a) Construct any vector \(\vec{a}\) that is pointing east.

(b) Construct a vector \(\vec{b}\) that is pointing west and has twice the magnitude of \(\vec{a}\).

(a)

(b)

(a)

(b)

(a) We are being asked to construct a vector \(\vec{a}\) of any magnitude whose direction is east.

One such vector is as shown below.

(b) As the magnitude of \(\vec{b}\) is twice that of \(\vec{a}\), the arrow representing \(\vec{b}\) will be twice as long as the arrow representing \(\vec{a}\).

As \(\vec{b}\) points in the opposite direction to \(\vec{a}\), the arrow representing \(\vec{b}\) points in the opposite direction to the arrow representing \(\vec{a}\).

Video Walkthroughs are currently in the process of being gradually uploaded, with that process expected to be finished by the end of October.

In the meantime, if you don't understand something in the Walkthrough above, feel free to reach out to Mr. Kenny on Teacher Chat!

Question 6

The length of a typical ruler is approximately \(30\mbox{ cm}\).

State the length of such a ruler:

(a) in millimetres

(b) in metres

(a) \(300\mbox{ mm}\)

(b) \(0.3\mbox{ m}\)

(a)

\begin{align}30\times \dfrac{1{,}000}{100}=300\mbox{ mm}\end{align}

(b)

\begin{align}30\times \dfrac{1}{100}=0.3\mbox{ m}\end{align}

(a) There are \(100\) centimetres in a metre and there are \(1{,}000\) millimetres in a metre.

Therefore, there are \(\dfrac{1{,}000}{100}=10\) millimetres in a centimetre. 

Hence, a ruler is \(30\times 10=300\) millimetres long.

(b) As there are \(100\) centimetres in a metre, there are \(\dfrac{1}{100}=0.01\) metres in a centimetre.

Therefore, a ruler is \(30\times 0.01=0.3\) metres long.

Video Walkthroughs are currently in the process of being gradually uploaded, with that process expected to be finished by the end of October.

In the meantime, if you don't understand something in the Walkthrough above, feel free to reach out to Mr. Kenny on Teacher Chat!

Question 7

Consider the following vector \(\vec{a}\).

(a) State the direction of this vector.

(b) Construct any two vectors pointing in the opposite direction to \(\vec{a}\).

(a) West (or to the left).

(b)

(a) West (or to the left).

(b)

(a) This vector is pointing west (or to the left).

(b) We are being asked to construct two vectors of any magnitude that are pointing east (or to the right), two of which are shown below.

Video Walkthroughs are currently in the process of being gradually uploaded, with that process expected to be finished by the end of October.

In the meantime, if you don't understand something in the Walkthrough above, feel free to reach out to Mr. Kenny on Teacher Chat!

Question 8

A train is travelling at a constand speed of \(30\mbox{ m/s}\).

Using a calculator, write this train’s speed in \(\mbox{km/h}\). 

\(108\mbox{ km/h}\)

\begin{align}30\mbox{ m/s}&= \frac{30 \mbox{ metres}}{1\mbox{ second}}\\&=\frac{30\times \frac{1}{1{,}000}\mbox{ kilometres}}{1\times \frac{1}{3{,}600}\mbox { hours}}\\&=\frac{\frac{30}{1{,}000}}{\frac{1}{3{,}600}}\frac{\mbox{kilometres}}{\mbox{hour}}\\&=108\mbox{ km/h}\end{align}

There are \(1{,}000\) metres in a kilometre. Therefore, there are \(\dfrac{1}{1{,}000}\) kilometres in a metre.

There are \(60\times 60=3{,}600\) seconds in an hour. Therefore, there are \(\dfrac{1}{3{,}600}\) hours in a second.

Hence, we can rewrite this train’s speed as follows:

\begin{align}30\mbox{ m/s}&= \frac{30 \mbox{ metres}}{1\mbox{ second}}\\&=\frac{30\times \frac{1}{1{,}000}\mbox{ kilometres}}{1\times \frac{1}{3{,}600}\mbox { hours}}\\&=\frac{\frac{30}{1{,}000}}{\frac{1}{3{,}600}}\frac{\mbox{kilometres}}{\mbox{hour}}\\&=108\mbox{ km/h}\end{align}

where the final answer was found using a calculator.

Video Walkthroughs are currently in the process of being gradually uploaded, with that process expected to be finished by the end of October.

In the meantime, if you don't understand something in the Walkthrough above, feel free to reach out to Mr. Kenny on Teacher Chat!

Question 9

Consider the following vector \(\vec{w}\).

(a) Redraw \(\vec{w}\).

(b) Construct a vector \(\vec{x}\) whose magnitude is half that of \(\vec{w}\) and that points in the opposite direction to \(\vec{w}\).

(c) Construct a vector \(\vec{y}\) whose magnitude is four times that of \(\vec{x}\) and that points in the opposite direction to \(\vec{x}\).

(d) What is the relationship between \(\vec{w}\) and \(\vec{y}\)? 

(a)

(b)

(c)

(d) \(\vec{y}\) has twice the magnitude of \(\vec{w}\) and both have the same direction.

(a)

(b)

(c)

(d) \(\vec{y}\) has twice the magnitude of \(\vec{w}\) and both have the same direction.

(a) This vector has been redrawn as below.

(b) As the magnitude of \(\vec{x}\) is half that of \(\vec{w}\), the arrow representing \(\vec{x}\) will be half as long as the arrow representing \(\vec{w}\).

As \(\vec{x}\) points in the opposite direction to \(\vec{w}\), the arrow representing \(\vec{x}\) points in the opposite direction to the arrow representing \(\vec{w}\).

(c) As the magnitude of \(\vec{y}\) is four times that of \(\vec{x}\), the arrow representing \(\vec{y}\) will be four times as long as the arrow representing \(\vec{x}\).

As \(\vec{y}\) points in the opposite direction to \(\vec{x}\), the arrow representing \(\vec{y}\) points in the opposite direction to the arrow representing \(\vec{x}\).

(d) As \(\vec{x}\) has half the magnitude of \(\vec{w}\), and as \(\vec{y}\) has four times the magnitude of \(\vec{x}\), \(\vec{y}\) has \(\dfrac{1}{2}\times 4=2\) times the magnitude of \(\vec{w}\).

Also, according to the above diagrams, \(\vec{y}\) is pointing in the same direction as \(\vec{w}\).

Video Walkthroughs are currently in the process of being gradually uploaded, with that process expected to be finished by the end of October.

In the meantime, if you don't understand something in the Walkthrough above, feel free to reach out to Mr. Kenny on Teacher Chat!

Question 10

Cement is being used to construct a house at rate of \(9\) kilograms per hour.

State this rate in grams per second.

\(2.5\mbox{ g/s}\)

\begin{align}9\mbox{ kg/h}&= \frac{9 \mbox{ kilograms}}{1\mbox{ hour}}\\&=\frac{9\times 1{,}000\mbox{ grams}}{1\times 3{,}600\mbox { seconds}}\\&=\frac{9{,}000}{3{,}600}\frac{\mbox{grams}}{\mbox{second}}\\&=2.5\mbox{ g/s}\end{align}

There are \(1{,}000\) grams in a kilogram and there are \(60\times 60=3{,}600\) seconds in an hour.

Therefore, we can rewrite this rate as follows:

\begin{align}9\mbox{ kg/h}&= \frac{9 \mbox{ kilograms}}{1\mbox{ hour}}\\&=\frac{9\times 1{,}000\mbox{ grams}}{1\times 3{,}600\mbox { seconds}}\\&=\frac{9{,}000}{3{,}600}\frac{\mbox{grams}}{\mbox{second}}\\&=2.5\mbox{ g/s}\end{align}

where the final answer was found using a calculator.

Video Walkthroughs are currently in the process of being gradually uploaded, with that process expected to be finished by the end of October.

In the meantime, if you don't understand something in the Walkthrough above, feel free to reach out to Mr. Kenny on Teacher Chat!

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