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VCE Stuff => VCE Mathematics => VCE Mathematics/Science/Technology => VCE Subjects + Help => VCE Specialist Mathematics => Topic started by: NE2000 on October 31, 2009, 01:47:32 pm

Title: minimum |r|
Post by: NE2000 on October 31, 2009, 01:47:32 pm: Under what conditions does the concept that the time of minimum |r| is given by when $\bold{r \cdot v} = 0$ apply?
Title: Re: minimum |r|
Post by: arthurk on October 31, 2009, 02:07:15 pm: Ah i see this as well although i usually just apply minimum distance formula this way seems much easier
Title: Re: minimum |r|
Post by: almostatrap on October 31, 2009, 02:24:15 pm: Quote from: arthurk on October 31, 2009, 02:07:15 pm
minimum distance formula

?
Title: Re: minimum |r|
Post by: kamil9876 on October 31, 2009, 02:40:06 pm: It can also be a maximum depending on how the curve bends. Generally, the maximum/minimum occurs either when $r.v=0$ or at the endpoints of motion(ie: where you start or finish). Completely analogous to the question of "when does f(x) is maximum when f'(x)=0 apply?".

In some simple cases it is always true such as "line closest to origin"(but make sure to check endpoints of motion just to be sure ;)) This can be derived from drawing a circle: http://vcenotes.com/forum/index.php/topic,15576.msg183111.html#msg183111

So I guess one way to be sure is by solving r.v=0 and checking which of those times (including the endpoints) gives the min/max distance. And this may or may not be simpler than solving $(|r|^2)'=0$

Here is an example I made up:

Consider the curve: $r(t)=(cos(t)+1)i + (sin(t)+1)j, 0 \leq t<2\pi$

When is the distance from origin min/max:
$<br />v(t)=-sin(t)i+cos(t)j$

$r.v=-cos(t)sin(t)-sin(t)+cos(t)sin(t)+cos(t)$

$0=cos(t)-sin(t)$

$tan(t)=1$

$t=\frac{\pi}{4}, \frac{3\pi}{4}$

Checking: first gives bigger distance, second gives gives smaller distance and $smaller<r(0)=r(2\pi)<bigger$ . Hence the answer follows.
Title: Re: minimum |r|
Post by: NE2000 on October 31, 2009, 03:03:51 pm: thanks for the explanation :)
Title: Re: minimum |r|
Post by: TrueTears on October 31, 2009, 03:57:42 pm: Quote from: almostatrap on October 31, 2009, 02:24:15 pm
Quote from: arthurk on October 31, 2009, 02:07:15 pm
minimum distance formula

?
He means find the minimum of the quadratic (or whatever expression it is) under the square root.
Title: Re: minimum |r|
Post by: GerrySly on October 31, 2009, 04:44:18 pm: Quote from: TrueTears on October 31, 2009, 03:57:42 pm
He means find the minimum of the quadratic (or whatever expression it is) under the square root.
How do you set that out (sorry for hijacking thread)? Like it says find the minimum distance between this and that vector or whatever, you find the distance in terms of something, then how do you set out the rest? I know I have to find the derivative of what's under the brackets and then check if it's a max/min but how do you set it out?

Just getting rid of the square root and differentiating never feels right, like I'm missing an explanatory step or something
Title: Re: minimum |r|
Post by: NE2000 on October 31, 2009, 05:28:55 pm: I just state "minimum is achieved when g(x) = enter what is under square root has a minimum"

"Hence let g'(x) = 0"
Title: Re: minimum |r|
Post by: arthurk on October 31, 2009, 05:37:12 pm: is that how uve previously done it NE2000?
it seems more like a methods technique though
Title: Re: minimum |r|
Post by: TrueTears on October 31, 2009, 06:40:28 pm: Quote from: GerrySly on October 31, 2009, 04:44:18 pm
Quote from: TrueTears on October 31, 2009, 03:57:42 pm
He means find the minimum of the quadratic (or whatever expression it is) under the square root.
How do you set that out (sorry for hijacking thread)? Like it says find the minimum distance between this and that vector or whatever, you find the distance in terms of something, then how do you set out the rest? I know I have to find the derivative of what's under the brackets and then check if it's a max/min but how do you set it out?

Just getting rid of the square root and differentiating never feels right, like I'm missing an explanatory step or something
Yeah NE2000 basically said it all :)
Title: Re: minimum |r|
Post by: kamil9876 on October 31, 2009, 09:48:27 pm: yeah I think that differentiating thing under the square root sign is just as good if not better in most cases.

Quote
Just getting rid of the square root and differentiating never feels right, like I'm missing an explanatory step or something

Seems pretty standard, but I think I even explained it once to someone here as since the function $f(x)=\sqrt{x}$ is always increasing the minimum value of f(x) occurs when the minimum value of x occurs.
Title: Re: minimum |r|
Post by: GerrySly on October 31, 2009, 09:52:53 pm: Quote from: kamil9876 on October 31, 2009, 09:48:27 pm
Seems pretty standard, but I think I even explained it once to someone here as since the function $f(x)=\sqrt{x}$ is always increasing the minimum value of f(x) occurs when the minimum value of x occurs.
Yeah I know it's pretty standard, just when they ask for the minimum value in other question my preceeding line is "Min occurs when dy/dx=0" then let dy/dx=0 and solve, just going from f(x)=sqrt(something) to d/dx(something) feels as if I could be explaining a step.

I'll just follow what NE2000 does, seems to be working fine :)