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VCE Stuff => VCE Mathematics => VCE Mathematics/Science/Technology => VCE Subjects + Help => VCE Specialist Mathematics => Topic started by: sachinmachin on September 02, 2009, 12:36:32 pm

Title: question for the true spesh genius!
Post by: sachinmachin on September 02, 2009, 12:36:32 pm: Part cii has got me stumped....rest is okay but relevant to that part
question 2
Let C = ai+bj and let R represent the position of an object which moves such that R.(R-C) = 0.
part a) Find, in terms of a and b, the cartesian equation of the path of the moving object.

part b) Describe the shape of the path and its key features,

part ci) show that R = (pcos(qt) + c)i + (psin(qt) + d)j represents the motion of a particle moving in a circle at constant speed, where p,q, c and d are constants.

??part cii)?? Hence find a vector equation to represent the motion of the object if C = 40i and the object is initially at C and travels at a constant speed of 3m/s in an anticlockwise direction.
Title: Re: question for the true spesh genius!
Post by: Mao on September 02, 2009, 01:34:32 pm: let $R = xi + yj,\; R - C = (x-a) i + (y-b) j$

$R\cdot (R-C) = x^2 - ax + y^2 - by = 0 \implies \left(x-\frac{a}{2}\right) ^2 + \left( y - \frac{b}{2} \right)^2 = \frac{a^2 + b^2}{4} = \left\| \frac{C}{2} \right\| ^2$

This is a circle centered the midpoint between origin and C with diameter of |C|. O and C are two ends of a diameter, and OCR makes a right-angled triangle (this is true for any circumscribed triangle that has the diameter as a side).

$R = \left( p \cos qt + \frac{a}{2} \right) i + \left(p \sin qt + \frac{b}{2} \right) j$ , where $p = \frac{\| C \|}{2}, q = \frac{2\pi}{T}$ (T is the period)

C has been specified to be 40i, hence $a = 40,\; b = 0,\; \| C \| = 40,\; p = 20$
The circumference is $\pi d = 40\pi$ , the period is the circumference divided by speed, which is also obtainable from q, hence $\frac{2\pi}{q} = \frac{40\pi}{3}\implies q = \frac{3}{20}$

Hence $R = (20\cos (3t/20) + 20) i + 20\sin (3t/20) j$
Title: Re: question for the true spesh genius!
Post by: sachinmachin on September 02, 2009, 01:51:11 pm: thanx a lot...
mao can do no wrong..
is this is an easy question though? in terms of vector calculus??
could i expect something like this on the exam??
Title: Re: question for the true spesh genius!
Post by: Mao on September 02, 2009, 01:56:28 pm: Yes you can.

This question isn't particularly easy, but it is fair.
Title: Re: question for the true spesh genius!
Post by: sachinmachin on September 02, 2009, 03:42:19 pm: one question Mao.
how does the value of C = 40. Would this not be considered as the radius rather than the diameter. as 40i runs along the the positive x-axis??
Title: Re: question for the true spesh genius!
Post by: Mao on September 02, 2009, 11:30:13 pm: Quote from: sachinmachin on September 02, 2009, 03:42:19 pm
one question Mao.
how does the value of C = 40. Would this not be considered as the radius rather than the diameter. as 40i runs along the the positive x-axis??

C is the diameter.
Title: Re: question for the true spesh genius!
Post by: believe_in_you on September 14, 2009, 07:28:50 pm: I dont get how you get b ! Can you help me pls ?

I am not good at Math at all