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Maths IA Help

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Hi,

Over the Christmas break, my maths teacher told us to come up with a "plan of attack" for our maths explorations. Our entire class isn't quite sure what that means so we're all at different stages of it. I was hoping to get some help with a few ideas that I have.

1. The Brachistochrone problem - I'm really interested in this problem but the problem I have with it is that I'm not sure if it is way out of the scope of the maths SL syllabus. All the IA's I've seen on this topic have been from HL so I know it isn't overdone but this has led me to wonder whether this lack of SL IAs on this topic is for a reason. From what I've seen this is primarily a question of calculus of variations and optimisation. 

For anyone who doesn't know, the brachistochrone problem was put forward by Johanne Bernoulli in 1696 and the problem goes like this: given two points A and B in a vertical plane, what is the curve traced out by a point acted on only by gravity, which starts at A and ends at B in the shortest time.

2. Minimising the drag force on a spacecraft - I have a lot of information on this but I haven't been able to find much specifically for IB maths. There is a bunch of stuff from NASA about this and I believe this is a question of calculus as well. 

Pretty sure I would go about this by finding the optimal size of a rocket to allow it to be as light as possible while carrying the most payload. When putting it like that it feels a bit simple. 

3. Finding the optimal orbital transfer - I know the optimal orbital transfer is a Hohmann transfer but I've always been interested in space flight and thought it would be cool to work my way there. This is also a question of calculus of variations I believe and I haven't been able to find much in terms of this for IB maths. 

4. Looking at Kepler's First and Second Laws - I know a bit about this from personal research and think it is really interesting. I was thinking of creating a star system with a planet orbiting around it, and then looking at how distance from the star affects the speed of said planet. I could using differentiation to see the relationship of this but other than that not really sure what else I could do.

Any input is greatly appreciated, please let me know your thoughts on these.

Thanks!

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1) You say that you are very interested in this topic, but make sure you can handle the maths involved. It seems to be quite a difficult topic for you if it was done only by HL students.  Also, make sure that you understand every single step of maths involved and don't just reproduce calculations from a website. Regurgitation of calculations does not make a well-scoring IA. You have to show understanding of the maths involved. Also, try and see if you can bring in other areas of maths into this.

2) This seems like a good topic. Its simple, and you say that you are interested in space travel, so you could possibly bring in that excitement you have. Don't worry if you can't relate it to the IB MATH SL syllabus. It can be from elsewhere. You just need to make sure you demonstrate an understanding of the math involved. The only problem I see is that you mention it is a "question of calculus". Just ensure that your IA doesn't turn into problem-solving. We do that in our exams.

3) Again don't worry if the math is beyond the syllabus, just ensure that you understand it.

4) I am not a physics person, but this seems advanced and requires a lot of effort. Don't want to put you off given how much interest you have in this topic. But you really have to brainstorm the calculations required and how you would go about it. If you can pull this off, it sounds amazing. But again, talk to your teacher about this one and find more information if you can. With how much you shared, there isn't much anyone call tell you.

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Adding to previous post

2) I wouldn't say this is simple. Drag force of an object is dependent upon its shape. It's near impossible to "calculate" the drag coefficient given an arbitrary shape and you need the coefficient for the force. 

4) Mathematical proofs to all three Kepler Laws should be available. Proof of II requires SL level of vectors and a tiny bit of calculus but essentially none. The other two proofs are algebraic. Some proofs are written for physics students and they may be mathematically less rigorous and be sure to modify them or at least acknowledge the fact. I like combining this with discussion of ellipses. On wikipedia I see Hohmann transfer is an application of Kepler's Third Law so best you do 4) first and if you have time maybe do a bit of 3).

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