What has both magnitude and direction?

XtinaVoiceWithin

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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Zhou Yu

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.

Ambitwistor

A hyperbola is not a rotated parabola.

Zhou Yu

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.

Integral

Staff Emeritus
Gold Member
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.

XtinaVoiceWithin

Thanks so much for putting so much effort to my questions. Gale

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...

Ambitwistor

Originally posted by Gale17
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...
I think I know what you're thinking, but you have it a little garbled.

First, by "oval" I'm guessing you mean an ellipse. 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.

So, we say that if you throw a ball, it follows a parabolic arc. But it's really following part of an elliptical arc that, over a comparatively small distance, is very similar to (but not the same as) a parabola.

Gale

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...

Ambitwistor

Originally posted by Gale17
heh... i actually sat for like 5 minutes trying to figure out the other word for oval...
Maybe it's "walltet".

Gale

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...

russ_watters

Mentor
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).

Ambitwistor

Originally posted by russ_watters
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).
Not really. It's true that parabolas and ellipses are conic sections with the properties you note. But the reason why a small arc of an ellipse looks like a parabola has nothing to do with the fact that they are both conic sections.

In fact, any trajectory due to any force law -- not just conic section solutions to an inverse square law -- will look like a parabola over small distances. This is a consequence of Taylor's theorem, expanding to second order. It arises because any gravitational field locally looks like a uniform field over a distance scale smaller than the scale of its gradient, and a parabola is the trajectory obtained from a uniform field.

Chi Meson

Homework Helper
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.

PhalanxGun

I Can Merely Assume You People Were Initially Referring To A "Hyperbola", Not A "Hyperbole", Which Is An Exaggeration. ;)

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