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The Directrix of a Parabola

https://isaacphysics.org/questions/directrix?board=a0609375-6803-40ea-8474-06dbe9f18a3c&stage=a_level

PART C - no idea where to even start - I initially thought the centre of the circle would stay constant but then realised I had the complete wrong idea.
My working for Part A and B:
IMG_6756.jpg
Any help would be greatly appreciated.
Original post by mosaurlodon
https://isaacphysics.org/questions/directrix?board=a0609375-6803-40ea-8474-06dbe9f18a3c&stage=a_level

PART C - no idea where to even start - I initially thought the centre of the circle would stay constant but then realised I had the complete wrong idea.


Some guides that would help.
Consider an arbitrary point say P(x,y) on the projectile path.
We know how to describe the x-coordinate and y-coordinate in terms of initial speed u, angle θ, g and t.
Use the trigo identity sin2θ + cos2θ = 1 to "transform x and y into circle equation" and we can find the centre of circle from the circle equation.
I am not sure what you mean - in fact I am not entirely sure whether I understand what the q is asking.
I picked the max height to be point P, and labelled desmos to what I thought was happening.

My initial thought was to use discriminant as both graphs meet at one point but not sure if this is correct approach and how sin2θ + cos2θ = 1 relates to this.
Reply 4
Original post by mosaurlodon
I am not sure what you mean - in fact I am not entirely sure whether I understand what the q is asking.
I picked the max height to be point P, and labelled desmos to what I thought was happening.

My initial thought was to use discriminant as both graphs meet at one point but not sure if this is correct approach and how sin2θ + cos2θ = 1 relates to this.
You want to show that at any time, the locus of (x,y) values for varying theta lie on a circle. So as above use the pythagoras identity c^2+s^2=1 and start from the original equations of motion
x = vtc
y = vts - gt^2/2
and rearrange them for s and c, square and sub and ...

tbh, you should almost be able to spot the radius and x and y centres without explicitly doing the rearrangement (x~rc, y~rs). As you should expect theyre functions of t and the radius will expand as t increases (it must start at zero) and the x centre is constant but the y centre decreases as time increases. You use the pythagorean identity to eliminate the parametric variable theta (as usual).

In desmos its something like
https://www.desmos.com/calculator/4hxhy96ctx
(edited 1 month ago)
Original post by mosaurlodon
I am not sure what you mean - in fact I am not entirely sure whether I understand what the q is asking.
I picked the max height to be point P, and labelled desmos to what I thought was happening.

My initial thought was to use discriminant as both graphs meet at one point but not sure if this is correct approach and how sin2θ + cos2θ = 1 relates to this.


If you want to know what the question is asking, see the animation in the link.
https://ibb.co/Lxj4Kkp

(edited 1 month ago)
Original post by mosaurlodon
I am not sure what you mean - in fact I am not entirely sure whether I understand what the q is asking. [\quote]

The guides (by right should be self-explanatory) because you have done it in part A.
Consider an arbitrary point say P(x,y) on the projectile path.
We know how to describe the x-coordinate and y-coordinate in terms of initial speed u, angle θ, g and t. (see below)
The Directrix of a Parabola01_crop.jpg
I say describe the x-coordinate in terms of initial speed u, angle θ, g and t means to express the x-coordinate in terms of initial speed u, angle θ, g and t.
Have you not done in Part A?
Use the trigo identity sin2θ + cos2θ = 1 to transform x and y into circle equation …(how)
Both x and y have a trigo ratio respectively, you can make the trigo ratio the subject and then square the trigo ratio to substitute them into the sin2θ + cos2θ = 1 and transform it into a circle equation:
(x-a)^2 + (y-b)^2 = r^2.

Advice:
Learn to read the sentences slowly and repeat them a few times slowly.
Very often, students like to rush through the sentences and refuse to read them again when they find the sentences confusing.
Original post by Eimmanuel
If you want to know what the question is asking, see the animation in the link.
https://ibb.co/Lxj4Kkp

Ohhhhhhh that makes sense to me now.
Actually what you're saying makes a lot more sense with that diagram - I in fact did not understand what the question was saying and had the complete wrong idea of the circle.
Ive managed to do both remaining parts now - thanks guys 😊

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