What has both magnitude and direction?

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Discussion Overview

The discussion revolves around the concepts of vectors and scalars, specifically what quantities possess both magnitude and direction versus those that have only magnitude. It also explores the nature of projectile motion, debating the path of projectiles and the geometric shapes associated with their trajectories, such as parabolas and hyperbolas.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants assert that quantities with both magnitude and direction are vectors, while those with only magnitude are scalars.
  • There is a consensus among some that the path of a projectile is a parabola, although others introduce the idea of hyperbolas and discuss their differences.
  • One participant elaborates on the mathematical definitions and properties of hyperbolas and parabolas, noting that hyperbolas have asymptotes and consist of two separate curves, while parabolas do not.
  • Another participant suggests that projectiles could be considered as following half-ovals due to the curvature of the Earth, leading to a discussion about the nature of projectile motion and its approximation as parabolic over small distances.
  • Ambitwistor mentions that while projectile motion is approximated as parabolic near the Earth's surface, it is ultimately elliptical, especially for longer distances or higher velocities.
  • There is a humorous exchange regarding the terminology used, with one participant clarifying the difference between "hyperbola" and "hyperbole."

Areas of Agreement / Disagreement

Participants generally agree that projectiles follow a parabolic path under certain conditions, but there is disagreement regarding the nature of this path over larger distances and the influence of Earth's curvature. Multiple competing views exist regarding the classification of projectile motion as either parabolic or elliptical.

Contextual Notes

Some statements rely on specific assumptions about the conditions under which projectile motion is analyzed, such as proximity to the Earth's surface and the effects of gravity. The discussion also highlights the potential confusion between similar geometric terms.

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1. What has both magnitude and direction?
2. What has only magnitude?
( is it scalar, vector, frame of reference)


3. What is the path of a projectile? (hyberbole or parabola)
Sorry if this question is kind of unclear.

Thanks !
 
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1. In relation to your first question about having magnitude and direction. The answer would be a vector.

2. If is only magnitude then it would be scalar.


3. When you say projectile, I am thinking something like a arrow being shot into the air and then travels in an arc of some kind.

In which case that would make it a Parabola.

A hyperbole is like a parabola except rotated 90 degrees.
 
I am still quite sure that parabola is the right answer.

However Ambitwistor, is right in saying that "A hyperbola is not a rotated parabola".

However I did say it is "like" a parabola, but having done a little more research into it, a parabola isn't really like a hyperbola at all.

____________________________________________________________

A hyperbola is the set of all points P(x,y) in the plane such that
| PF1 - PF2 | = 2a

Both F1 and F2 are focus points, and the difference between them is always the same.

A hyperbola also has asymptotes which are the boundaries of the hyperbola. A parabola however does not have such boundaries.

Also a Hyperbola is where there are two curves, and F1 - F2 will always equal a constant of 2a.

A Parabola only consists of 1 line.

____________________________________________________________

I hope this compensates a little for my mistake about the difference between a parabola and a hyperbola.
 
Both the parabola and the hyperbola are conic sections. The parabola is described by

y = (x-h)2+k

This is arc of a projectile, it is created by slicing a cone by plane not parallel to the axis but intersecting the base of the cone.

A hyperbola is described by

(x-h)2/a+ (y-k)2/b= 1


This is created by slicing a cone in a plane parallel to the main axis. It is charaterized by 2 non connected pieces.

This is the path followed by a body moving in space which passes near a massive body but is not in an orbit around the body.
 
Thanks so much for putting so much effort to my questions.:smile:
 
wait, i thought i read in here a while ago that projectiles were half ovals... is that right? i mean, i know we all just do the math as if it were a parabola, but its actually half an oval because of the curvature of the Earth right? or did i totally just make that up...
 
yeah thanks Ambitwistor that's it... heh... i actually sat for like 5 minutes trying to figure out the other word for oval... but its late, didn't come to me... but yeah, i knew they were close anyways so it didn't really matter... but i was just making sure that i hadn't made something up...
 
Maybe it's "walltet".

OOOOOH! hahaha... you're a clever one aren't you... yeah... so I'm not so great with english... pfft, its only my native language...
 
Originally posted by Ambitwistor
It's true that if you throw an object, it follows an elliptical orbit (if you throw it at less than escape velocity). A partial arc of an ellipse is not a parabola. (i.e., if you cut a piece off an ellipse, the piece is never a parabola.) However, over a small distance, an arc of an ellipse is well approximated by a parabola -- they're very similar in shape.
Another way to think of it is circles and parabolas are special cases of ellipses. A circle is an ellipse with zero distance between the foci and a parabola is an ellipse with infinite distance between the foci. That is why over small distances a parabola and ellipse are very close - they are siblings (conic sections).
 
  • #10
Projectile motion is always, ultimately, elliptical. FOr the motion to be truly parabolic, the acceleration due to gravity would have to always be in one direction. THis condition is only approximated when we are near the Earth's surface because we cannot detect the change in direction of "g". Over relatively small distances we use parabolic projectile motion. FOr intercontinental ballistic missiles, however, elliptical projectile motion is followed (plus coriolis effect, air resistance, and other fun stuff, and of course for satellite motion, it's totally elliptical.
 
  • #11
I Can Merely Assume You People Were Initially Referring To A "Hyperbola", Not A "Hyperbole", Which Is An Exaggeration. ;)
 

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