HSC 2013 MX2 Marathon (archive) (2 Viewers)

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nightweaver066

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Re: HSC 2013 4U Marathon

There is a contradiction in your working.
I don't think it's a contradiction, i just made a mistake in the 5th line so i technically didn't prove the left side of the inequality :(
 

Sy123

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Re: HSC 2013 4U Marathon

Isn't it not the other way around? idk O_O

The 'proof' that

doesn't make sense either - e is just a constant, it shouldn't matter what n is doing! (I think it's a directional issue here)
Apologies, it should be the other way around. Fixed now.
 

Sy123

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Re: HSC 2013 4U Marathon









 
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y=|sinx|?


Edit: dunno if that's the right motion equation since I'm not sure how to incorporate the fact that 1 revolution = 1 second
 
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Sy123

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Re: HSC 2013 4U Marathon

y=|sinx|?


Edit: dunno if that's the right motion equation since I'm not sure how to incorporate the fact that 1 revolution = 1 second
Not quite heh, I made it a show that to compare solutions
 

RealiseNothing

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Re: HSC 2013 4U Marathon

y=|sinx|?


Edit: dunno if that's the right motion equation since I'm not sure how to incorporate the fact that 1 revolution = 1 second
I'm pretty sure the speed the circle turns is irrelevant as it would still trace the same path.
 

Sy123

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Re: HSC 2013 4U Marathon

I'm pretty sure the speed the circle turns is irrelevant as it would still trace the same path.
This is true, I gave it to sort of make the elimination of variables easier. Because its easier to imagine it if you are given a time

But reason why this is true is:

 

Sy123

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Re: HSC 2013 4U Marathon

Yeah, my answer was wrong since I messed up the initial conditions, it should be correct now.
 

Sy123

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Re: HSC 2013 4U Marathon

Here is my solution to the question.

==============

Ok, so we need to model the moving marker, and I know I can simplify the problem by finding the parametric equations of the red marker.
So I need to find the parametric equations of the marker, but since this red marker is dependent on the circle. I will first investigate the circle to see what I can find:

Establish the equation of the circle:



For some variable centre (p,1).

Now I know that the parametric equations of this circle is:




These equations are able to give me any point I want on that circle, and the point we want to model or represent is the red marker.
So we need to find a relationship between p and theta such that they model the red marker.

This is why the time variable is useful to imagine the following situation. Ok, so our circle makes 1 revolution per second, but its rolling, hence the distance the centre covers in 1 second, is equal to the circumference (draw this out if it doesn't make sense)

Hence,
Now we need to find a way to model theta in terms of t. This is easy enough anyway, every second theta increases by 2pi, hence



However we need to be careful here, theta is actually decreasing, as in it is increasing in the negative direction since our theta is going clockwise (as stated in the question), hence indeed it comes about that:



Recall the initial condition for the red marker being (0,0) When the circle has an equation of
However when theta is zero initially, our inital conditions become (1,1). To make our inital conditions (0,0) we must take away pi/2 to theta (imagine the unit circle of trigonometry, similar logic is involved).

Hence our new equations are:





Simplification





Now we have the parametric equations, we now need to isolate one and form the cartesian equation:





Use the right angle triangle in order to simplify the second term

 

Sy123

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Re: HSC 2013 4U Marathon

Here is a nice result I proved:





 
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Sy123

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Re: HSC 2013 4U Marathon

A neat result that I proved:




















 
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