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Circles, Centre, parabolas, motion

Come up to a bit of a strange question here and I’m stuck. The first section seems focused on the ‘Directrix of a Parabola’. However the part I’m stuck on it doesn’t seem necessary to use/understand that anyway.

Question here: 7409751C-7694-439E-A80F-7DD963CEF4E0.jpg.jpeg

I started of with this E97F6026-0CCD-41C1-B99F-21D37AD9674B.jpg.jpeg following the hint 2C3A0F92-BCDF-47CA-AE49-219AB2098C43.jpg.jpeg

But I’m not sure what to do next? Assuming I’ve got that correct then I’m not sure what I could do to make theta independent of the distance (radius) unless v or t was equal to zero…

Would really appreciate help with this one!

More (higher resolution?) pictures of the question and the other parts here: https://m.imgur.com/a/4VTHSvw
3636B2BA-356D-41FF-872E-54575E37FEBA.jpg.jpeg50.6KB
Sorry I can't help (far too rusty and haven't done this sort of stuff since circa 1982!). But what a beautiful bit of maths!
Original post by Maximus 190
Come up to a bit of a strange question here and I’m stuck. The first section seems focused on the ‘Directrix of a Parabola’. However the part I’m stuck on it doesn’t seem necessary to use/understand that anyway.

Question here: 7409751C-7694-439E-A80F-7DD963CEF4E0.jpg.jpeg

I started of with this E97F6026-0CCD-41C1-B99F-21D37AD9674B.jpg.jpeg following the hint 2C3A0F92-BCDF-47CA-AE49-219AB2098C43.jpg.jpeg

But I’m not sure what to do next? Assuming I’ve got that correct then I’m not sure what I could do to make theta independent of the distance (radius) unless v or t was equal to zero…

Would really appreciate help with this one!

More (higher resolution?) pictures of the question and the other parts here: https://m.imgur.com/a/4VTHSvw


Although he's deleted it, @mqb2766 was on the right lines.

If you consider a particle going straight up and one going straight down, IF they lie on a circle, those two particle's positions must form the extermities of a diameter and their midpoint must be the centre of the circle. Hence, if it is a circle, the centre of circle must be ....

Then continue with your Pythagoras, and theta will be eliminated.... showing it is a circle (dependent on t).
(edited 4 years ago)
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Avatar for mqb2766
mqb2766
21
Original post by ghostwalker
Although he's deleted it, @mqb2766 was on the right lines.

If you consider a particle going straight up and one going straight down, IF they lie on a circle, those two particle's positions must form the extermities of a diameter and their midpoint must be the centre of the circle. Hence, if it is a circle, the centre of circle must be ....

Then continue with your Pythagoras, and theta will be eliminated.... showing it is a circle (dependent on t).


Yeah, thought it was ~1/2 right but was trying to sort out the 1/2 wrong bit :-)
Original post by ghostwalker
Although he's deleted it, @mqb2766 was on the right lines.

If you consider a particle going straight up and one going straight down, IF they lie on a circle, those two particle's positions must form the extermities of a diameter and their midpoint must be the centre of the circle. Hence, if it is a circle, the centre of circle must be ....

Then continue with your Pythagoras, and theta will be eliminated.... showing it is a circle (dependent on t).


Okay I have found the centre and radius by assuming it is a circle and using the fact that the particles going up and down are opposite ends, midpoint is centre etc. This gives me the radius as a function of time = vt which is correct. However, when proving it is a circle independent of theta I don’t see what you mean my continue with my Pythagoras. Seems like a huge mess. 76C4D37F-0FB6-48A0-8289-103655C4FC90.jpg.jpeg
Original post by Maximus 190
Okay I have found the centre and radius by assuming it is a circle and using the fact that the particles going up and down are opposite ends, midpoint is centre etc. This gives me the radius as a function of time = vt which is correct. However, when proving it is a circle independent of theta I don’t see what you mean my continue with my Pythagoras. Seems like a huge mess. 76C4D37F-0FB6-48A0-8289-103655C4FC90.jpg.jpeg



You've worked out where the centre must be. So, why didn't you just replace c in your original Pythagoras. Everything simplifies and theta vanishes.
Original post by ghostwalker
You've worked out where the centre must be. So, why didn't you just replace c in your original Pythagoras. Everything simplifies and theta vanishes.


Oh yeah, very cool! Thanks :smile:
Out of interest what was the final answer for the y-coord (as a function of time)?
(edited 3 years ago)

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