Thinking [email protected] #3

JC Mathematics, Mathematics

[email protected] is a series of questions that we, as tutors feel that are useful in helping students think and improve their understanding.

Thinking [email protected] is curated by KS. More of him can be found here.

This is a question from 1976 A’levels Paper 2. I thought it is pretty interesting to discuss the question with a little extension.

(a) In how many ways can 5 copies of a book be distributed among 10 people, if no-one gets more than one copy?

(b) In how many ways can 5 different books be distributed among 10 people if each person can get any number of books?

So now, let us modify it a bit.

(c) In how many ways can 5 copies of a book be distributed among 10 people if each person can get any number of books?

Notice that the difference between (b) and (c) is that the book distributed is not identical. So for (c), we are pretty much distributing r identical balls to n distinct boxes. Whereas for (b) , we are pretty much distributing r distinct balls to n distinct boxes.

Thinking [email protected] #2

JC Physics

[email protected] is a series of questions that we, as tutors feel that are useful in helping students think and improve their understanding.

Thinking [email protected] is curated by Aaron. More of him can be found here.

Assuming an object is moving in a circular motion in a polar coordinate system given by x = r \text{cos} \theta and y = r \text{sin} \theta. Can you derive the formula of the centripetal acceleration? Hint: look at one of the axes and think of the direction and what is centripetal acceleration

Thinking [email protected] #1

JC Physics

[email protected] is a series of questions that we, as tutors feel that are useful in helping students think and improve their understanding.

Thinking [email protected] is curated by Aaron. More of him can be found here.

Kepler’s Third Law states that the period of a planet’s orbit is directly proportional to the cube of the semi-major axis of its orbit. This allows scientists to calculate distances of planets in our solar system. Can you derive the equation using what you have learned in Motion in a Circle and Gravitational Field?

Physics@TheCulture #1

[email protected] #1

JC Physics

[email protected] is a series of questions that we, as tutors feel that are useful in helping students think and improve their understanding.

Thinking [email protected] is curated by Aaron. More of him can be found here.

Why study Physics?

I have questioned many students and getting answers ranging from “Just a subject that I have to take in A-Level” to “I do not know what else to take?”

It is not just a subject:

Physics can be seen as a ‘tool’ to enable you to be more logical and you practise and learn to question ‘Why’, which is an invaluable asset to many other jobs or careers

A student’s dad was having a good chat with me and he was a control systems engineer back in his younger years in The States for a good decade or so. Thereafter he went on to take an MBA and he is now an MD in a local bank. Even if you are not going to major in engineering, it can be a good training.

Helping out with the house:

You can of course repair and fix things in the house without studying Physics but if you know more about topics like pressure, forces, electricity and how things work, won’t it be awesome that you are more knowledgeable? It can be a general knowledge and it will help you out in life along the way when things are screwed at home.

It puts your maths to good use:

Lots of formulas? Well, they help physicists and engineers to understand the world we live in.

Becoming an Engineer:

It is a myth that only bankers are being paid well. Only the good bankers are well paid, similar to engineers. If you excel in the things you do, you will be rewarded. Engineers are not technicians. Yes, they can fix things but they innovate and SOLVE problems too.

A world without Physics:

You will not be looking at this now on your computer/laptop or mobile device literally. Engineers continuously apply Physics to push the frontiers of technology.

It is not just a subject:

Physics can be seen as a ‘tool’ to enable you to be more logical and you practise and learn to question ‘Why’, which is an invaluable asset to many other jobs or careers

A student’s dad was having a good chat with me and he was a control systems engineer back in his younger years in The States for a good decade or so. Thereafter he went on to take an MBA and he is now an MD in a local bank. Even if you are not going to major in engineering, it can be a good training.

Helping out with the house:

You can of course repair and fix things in the house without studying Physics but if you know more about topics like pressure, forces, electricity and how things work, won’t it be awesome that you are more knowledgeable? It can be a general knowledge and it will help you out in life along the way when things are screwed at home.

It puts your maths to good use:

Lots of formulas? Well, they help physicists and engineers to understand the world we live in.

Becoming an Engineer:

It is a myth that only bankers are being paid well. Only the good bankers are well paid, similar to engineers. If you excel in the things you do, you will be rewarded. Engineers are not technicians. Yes, they can fix things but they innovate and SOLVE problems too.

A world without Physics:

You will not be looking at this now on your computer/laptop or mobile device literally. Engineers continuously apply Physics to push the frontiers of technology.

Vectors Question #4

Vectors Question #4

JC Mathematics

Another interesting vectors question.

The fixed point A has position vector a relative to a fixed point O. A variable point P has position vector r relative to O. Find the locus of P if r \bullet (ra) = 0.

Vectors Question #3

Vectors Question #3

JC Mathematics

This is a question a student sent me a few days back, and I shared with my class.

Find the Cartesian equation of the locus of all points (plane) that is equidistant of the xy plane and xz plane.

The following should aid students to visualise.

xy-, xz-, yz-planes

Sidenote: I think Vectors is a very important topic for 9758 as its applications are wide. Students should do their best to understand the topic. I will share a few more applied questions next week when I have time.

A little reminder to students doing Calculus now

A little reminder to students doing Calculus now

JC Mathematics

When \frac{dy}{dx} = 0, it implies we have a stationary point.

To determine the nature of the stationary point, we can do either the first derivative test or the second derivative.

The first derivative test:

First Derivative Test

Students should write the actual values of \alpha^-, \alpha, \alpha^+ and \frac{dy}{dx} in the table.

We use this under these two situations:
1. \frac{d^2y}{dx^2} is difficult to solve for, that is, \frac{dy}{dx} is tough to be differentiated
2. \frac{d^2y}{dx^2} = 0

The second derivative test:

Second Derivative Test

Other things students should take note is concavity and drawing of the derivative graph.

Responsibility of Protecting the Environment

Responsibility of Protecting the Environment

JC General Paper

Environment is a common and easy essay question that could potentially come out for the exams… Students should always analyze their essays from a stakeholder’s perspective to give a more holistic evaluation.

Developed countries

Governments in developed countries have the necessary skills, influence and resources to aid the protection of the environment. They would have the financial means to invest in necessary science and technology which could improve the environment. At the same time, they would also possess political clout to enact policies to force companies to comply with their environmental standards. Education and campaigning to change the mindset of the people towards the environment would be within their means as people are generally more educated and aware.

Businesses could practise corporate social responsibility, and stand to gain since green consumers are more likely to purchase their products. Consumers are more concerned with the origins of their products they buy today. As such, MNCs could influence these green consumers to raise awareness about environmental issues using their products.

Individuals could make a difference by starting with themselves first. A small step could go a long way such as supporting Earth Hour or even car pooling.

NGOs no doubt would have more political clout than individuals, and do not have the same restrictions that are placed upon government agencies. They do not have to balance certain commitments such as the living standards in the country. This would result in them having greater flexibility when it comes to their operations and should be able to achieve more. \

Developing countries

Governments have the sole responsibility for the enactment and enforcement of policies to protect the environment. Unfortunately, their main priority is to ensure economic growth rather than the protection of the environment. Developing countries are more likely to have unstable government and corruption is usually rampant making it difficult to care for the environment. E.g Shell engaged in dodgy/shady dealings with the government officials in Nigeria in order to gain a foothold in the country.

Businesses are usually in the primary and secondary sector which focus on the harvesting of raw materials and manufacturing. These are highly damaging to the environment as the mining process results in large amounts of pollutants and waste. Many of their factories are owned by MNCs, and they would have little say in the production methods used.

Individuals concern would be to earn sufficient money to stay alive, they would not mind harnessing the earth for the raw materials or resorting to more efficient methods of clearing the land if they could help to save some money for them.

Considering the situation in both developed and developing countries, who do you think should be responsible for the environment?