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Monika M.
Teacher with a prior major in physics
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Physics (Waves and Optics)
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Question:

The Sun is our source of warmth and it keeps Earth's temperature habitable. Your friend has heard that space is a vacuum and that there is nothing there. They have also heard that light is a wave. They are perplexed as to how the heat from the Sun can then get to Earth as there is nothing in space for the waves to travel through. Explain to your friend how the energy from the Sun get to Earth through the vacuum of space.

Monika M.
Answer:

The introduction of your friend in the question shows that a more informal answer is desired. This is to keep you from giving an answer from memory and to rather show that you understand the phenomenon in question. In general it is best to always rather understand things in physics than memorising it. Your answer should be something along the lines of: Well it is actually easy to understand my friend. The Sun, as you know, gives off light. Light itself has energy inside of it. When there is a large light source like a fire for a barbecue you can feel the heat of it. That is the light's energy being carried over to your skin. Something unique about light is that even though it is a wave it does not require any medium to travel. So the fact that space is a vacuum makes no bit of a difference for the light and it will still bring warmth to our planet. The main things that are needed in this answer is that light is self-propagating and that light carries energy from one point to another. Using an example of everyday things will only help to show that you understand the given phenomenon.

Physics (Electricity and Magnetism)
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Question:

Two charged spheres A and B are suspended with a piece of non conductive material and held in place. Sphere A has a charge of 5C. Sphere B has a charge of -3C. The two spheres are released. 1. What series of events will you observe if the spheres are released? 2. What will the charge be of sphere A at the end of the series of events in question 1.

Monika M.
Answer:

The first question is testing if you understand how charges behave. The second question requires you to do a calculation based on your answer in question 1. 1. Quite often you will not be given a drawing. If you do not have a drawing it is best to draw the problem yourself just to help make sure you understand the problem correctly. The answer for question 1 will be something along the lines of: Sphere A and B are oppositely charged and will therefore be attracted to each other. Once they touch each other the electrons from sphere B will be transferred to sphere A until they are equally charged. Since the positive charge is the bigger charge they will both then be positively charged and will repel each other. The 3 main parts in this that you must state is that they will initially attract one another; that charge will be transferred on contact until they are equally charged; and that the resulting charge will drive them apart again. (Note: Remember, like charges repel and opposite charges attract) 2. Keeping in mind that the spheres will have the exact same charge after they touched each other we will just have to calculate the average charge of the two spheres. Your textbook will quite likely have given you the following equation: $$(q_1 + q_2)/2$$ If you look close at it you will notice that it is simply calculating the average. So what we will do is add both charges together and then divide by 2. $$(5 - 3)/2$$ $$ = 1C$$ (Note: Remember to make sure that you write in your units else it can be anything ranging from Coulomb to bears.) Sometimes you will get a mark for using the correct formula; so it is best to write out the entire solution as below: $$(q_1 + q_2)/2$$ $$= (5 - 3)/2$$ $$ = 1C$$

Algebra
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Question:

Fully factorise the following expression: $$4x^2 + 10xy + 6y^2$$

Monika M.
Answer:

There are two main ways to solve this problem. You can either directly factorise the trinomial or rather look for a common factor in order to simplify the expression and then factoris the new trinomial. I personally prefer to look for common factors first as it will make the following trinomial factorisation easier. Step 1: Find the common factor. Looking at each of the terms you will see that all of the numbers can be divided by 2. We can do nothing with the letter. We will therefore "take the 2 out of the expression". It is important to remember that there is no = sign and therefore you can't throw the 2 away. When we take the two out of the equation it will instead be written before a bracket and the rest of the expression will be written inside of the bracket. $$2(2x^2 + 5xy + 3y^2)$$ To test if this is correctly done we can multiply the 2 back in and compare the answer with our starting expression. If you often forget minus signs or other little mistakes I would recommend too always check each step. It is also good practice and will help you improve your math. Step 2: Factorise the trinomial inside of the bracket. You can either remove the trinomial and solve it on its own and then reintroduce it back into the expression; or you can solve it as is. I will do both methods below. I would recommend to solve the problem as is. Factorising a trinomial: Find the factors of term one and term three. For the expression $$2(2x^2 + 5xy + 3y^2)$$ we have that: Term one's factors are: 2 × 1 Term three's factors are: 3 × 1 For the expression $$4x^2 + 10xy + 6y^2$$ we have that: Term one's factors are: 4 × 1 and 2 × 2 Term three's factors are: 6 × 1 and 3 ×2 Next we need to find a combination of the factors of term one and term three that will result in term two's numerical value. For the expression $$2(2x^2 + 5xy + 3y^2)$$ we have that: 2 × 1 + 3 × 1 = 5 and 2 × 3 + 1 × 1 = 7 The first combination is the correct one. For the expression $$4x^2 + 10xy + 6y^2$$ we have that: 4 × 1 + 6 × 1 = 10 ; 4 × 6 + 1 × 1 = 25 ; 2 × 6 + 2 × 1 = 14 ; 4 × 3 + 1 × 2 = 14 ... The first combination is the correct one. Next we organise the numbers correctly in the brackets that we will now write. There are only plus signs, therefore there will only be plus signs in our brackets. (This is also why we only added the multiplication combinations above.) So for expression $$2(2x^2 + 5xy + 3y^2)$$ we will have the following: $$2(2x + 3y)(x + y)$$ So for expression $$4x^2 + 10xy + 6y^2$$ we will have the following: $$(4x + 6y)(x + y)$$ What you will notice is that if you multiply the 2 into the firsts bracket you will have the exact same answer you would get if you did not take out the common factor. (You can check your answer by multiplying out your answer. If it is the same as the starting expression then you are correct.) Below I will show you how to write all of this without the comments. Notation in mathematics is very important so do make sure your notation is correct. Notation when using the common factor method: $$4x^2 + 10xy + 6y^2$$ $$ = 2(2x^2 + 5xy + 3y^2)$$ $$ = 2(2x + 3y)(x + y)$$ Notation when using the common factor method and taking the bracket away: $$4x^2 + 10xy + 6y^2$$ $$ = 2(2x^2 + 5xy + 3y^2)$$ $$2x^2 + 5xy + 3y^2$$ $$ = (2x + 3y)(x + y)$$ $$ => 2(2x^2 + 5xy + 3y^2)$$ $$ = 2(2x + 3y)(x + y)$$ (This method of solving a segment at a time can be useful where you have to simplify a fraction through factorisation.) Notation when factorising the trinomial directly: $$4x^2 + 10xy + 6y^2$$ $$ = (4x + 6y)(x + y)$$ Note: You are generally permitted to write out the factors of term 1 and 3 on the side to help you keep track of the calculations. If your teacher is okay with you doing this, then I would recommend that you always do it. It may take a bit of your time, but it will make it easier for you to solve the problem.

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