View Full Version : acceleration due to gravity

thkaufm

2002-Feb-01, 06:01 PM

If an object in freefall accelerates due to gravity, how does something achieve orbit?

How can an object's sideways motion ever balance out it's falling motion if it's always falling increasingly faster?

ToSeek

2002-Feb-01, 06:12 PM

On 2002-02-01 13:01, thkaufm wrote:

If an object in freefall accelerates due to gravity, how does something achieve orbit?

How can an object's sideways motion ever balance out it's falling motion if it's always falling increasingly faster?

If the world were flat, you'd be right. However, the world is round, so the ground slips out from underneath as the object is falling. At the right speed, the object never catches up.

Try this cool applet (http://www.phys.virginia.edu/classes/109N/more_stuff/Applets/newt/newtmtn.html) for a demonstration, if your browser can handle Java.

thkaufm

2002-Feb-01, 08:28 PM

If the world were flat, you'd be right. However, the world is round, so the ground slips out from underneath as the object is falling. At the right speed, the object never catches up.

I understand that part, but what is the right speed? An object accelerates as it falls so why doesn't the ground have to slip out from underneath at an ever increasing speed.

In other words, why does an oject accelerate in gravity, is it because it is getting closer to the source. An object in orbit never gets any closer to the source therefor it doesn't fall any faster?

Tom

Hale_Bopp

2002-Feb-01, 08:42 PM

There are two ways an object can accelerate : by changing its speed or chaning its direction of travel. An object in orbit actually maintains a constant speed. The acceleration it expereinces, the good old 9.8m/s^2, is due to its change in direction.

Does that help at all?

Rob

GrapesOfWrath

2002-Feb-01, 08:57 PM

On 2002-02-01 15:28, thkaufm wrote:

In other words, why does an oject accelerate in gravity, is it because it is getting closer to the source. An object in orbit never gets any closer to the source therefor it doesn't fall any faster?

No, Tom, that's not it. The amount of acceleration does depend upon how close you are, but that is not why it accelerates. Let's see, I did this in the shower the other day...

If you have an orbital velocity of v, at a radius of r, then the satellite is turning through an angle of v x t / r in time t. The difference in the velocity vectors from one time to time t will just be that angle. To get the actual value, multiply the angle times v, or v^2 x t / r, and you divide that by t to get the acceleration. So, a satellite moving in a circle at radius r has to accelerate at v^2/r.

The gravity at radius r of a mass M is just GM/r^2, Newton's equation. So, if v^2/r equals GM/r^2, then the satellite will move in a perfect circle around the mass M. You can solve for v, and you find that

v = sqrt(GM/r)

ToSeek

2002-Feb-01, 09:04 PM

On 2002-02-01 15:28, thkaufm wrote:

I understand that part, but what is the right speed? An object accelerates as it falls so why doesn't the ground have to slip out from underneath at an ever increasing speed.

Well, I think another way of looking at it is that gravity makes the object DECELERATE in the direction it's currently going and accelerate in a new direction. If this is balanced, it will be in a circular orbit. If it's a little off, it will be elliptical. If it's way off, the object will either go thud or go flying off.

GrapesOfWrath

2002-Feb-01, 10:59 PM

On 2002-02-01 16:04, ToSeek wrote:

Well, I think another way of looking at it is that gravity makes the object DECELERATE in the direction it's currently going and accelerate in a new direction.

I gotta disagree with that one. The force of gravity is perpendicular to its velocity, so it cannot decelerate it in that direction.

Hale_Bopp

2002-Feb-02, 03:47 AM

I see what ToSeek is getting at. Let's look at the Earth from above the north pole and set up an x-y axis. At an initial time t= 0, let's say the object has a velocity of c in the +x direction. As the object orbits, 1/4 of an orbit later, its velocity will be c in the -y direction (counterclockwise orbit). From this perspective, its velocity in the x-direction has decreased and it has gained a negative velocity in the y direction. After half an orbit, It's velocity in the y direction is 0 and its velocity in the x direction is c. After 3/4 orbit, its velocity in the x direction is 0 and its velocity in the y direction is c. Then we are back to the beginning.

This problem is much simpler in radial coordinates. The acceleration is in the r direction at 9.8m/s^2 and its velocity is always in the theta direction and is constant!

Wish I knew all the code for the math symbols...would have made this more coherent.

Rob

ToSeek

2002-Feb-02, 03:21 PM

On 2002-02-01 17:59, GrapesOfWrath wrote:

On 2002-02-01 16:04, ToSeek wrote:

Well, I think another way of looking at it is that gravity makes the object DECELERATE in the direction it's currently going and accelerate in a new direction.

I gotta disagree with that one. The force of gravity is perpendicular to its velocity, so it cannot decelerate it in that direction.

Two comments:

1. You're right, but only for a perfectly circular orbit. If the orbit is at all elliptical, then gravity will be accelerating or decelerating it at all points except perigee and apogee.

2. Even in a perfectly circular orbit, if you break the velocity vector at a point into horizontal and vertical components and keep the same coordinate frame in space (not an Earth-centered frame), the horizontal velocity will be reduced as the vertical velocity increases, until the horizontal component is all of it, 1/4 of the way around (as Hale_Bopp indicates).

ToSeek

2002-Feb-02, 03:22 PM

After re-reading Hale_Bopp, I think we're saying almost exactly the same thing. Oh, well.

thkaufm

2002-Feb-02, 03:43 PM

It makes sense now that I think about it using the fixed coordinate system point of view.

Tom

GrapesOfWrath

2002-Feb-02, 05:02 PM

On 2002-02-02 10:21, ToSeek wrote:

1. You're right, but only for a perfectly circular orbit. If the orbit is at all elliptical, then gravity will be accelerating or decelerating it at all points except perigee and apogee.

No argument there, but the example of circular orbits should give you pause.

Another way to think about it is an analogy with a tossed ball. If you were to just drop a ball through the earth (I think we've had this discussion many times), it would pick up speed until the center of the earth. Its momentum would carry it to the other side, and the motion would start over.

If you toss a ball into the air at an angle, it will have two components--one which moves it forward, the other up. The up component is opposed by gravity, and it will eventually decay, and then reverse itself. The result is a parabola, as we all know.

If you were to throw a ball perpendicular to the earth, at orbital velocity, it would similarly follow an arch. The gravity would pull it towards the earth (same as "dropping" the ball), giving it its "forward" movement. And, sure enough, the period of such a ball (or satellite) is the same whether you drop the ball, or toss it into orbit. It takes the same amount of time to orbit the earth as it would to fall straight through it and back.

GENIUS'02

2002-Feb-04, 12:51 AM

The acceleration it expereinces, the good old 9.8m/s^2, is due to its change in direction.

sorry thats not actually true the accelleration due to gravity for an object in orbit is not the good old 9.8ms^-2 in stead it can be worked out using this formular:

g = G (mass{of earth})/the radius of the orbit squared

G=6.67 *10^-11

David Simmons

2002-Feb-04, 02:11 AM

On 2002-02-02 12:02, GrapesOfWrath wrote:

If you toss a ball into the air at an angle, it will have two components--one which moves it forward, the other up. The up component is opposed by gravity, and it will eventually decay, and then reverse itself. The result is a parabola, as we all know.

(nitpick) Actually, neglecting air resistance, the path is part of a long narrow ellipse. It would only be a parabola if the direction of the force of gravity were always parallel. But gravity always acts toward a point at the center so the direction of the force is always changing along the path of the ball. (/nitpick)

There, I'm almost ashamed of myself, but did that stop me? Nooooo.

<font size=-1>[ This Message was edited by: David Simmons on 2002-02-03 21:12 ]</font>

GrapesOfWrath

2002-Feb-04, 08:03 AM

On 2002-02-03 21:11, David Simmons wrote:

Actually, neglecting air resistance, the path is part of a long narrow ellipse. It would only be a parabola if the direction of the force of gravity were always parallel. But gravity always acts toward a point at the center so the direction of the force is always changing along the path of the ball. (/nitpick)

Great nitpick! Of course, though, gravity doesn't always act towards a point at the center. Even without the coriolis effect (that is, if it weren't rotating), the gravity on the oblate spheroid of the Earth would point away from the center of the Earth by kilometers. Local deviations (mountains, buried high-density masses) modify it even more.

There, I'm almost ashamed of myself, but did that stop me? Nooooo.

One good nitpick deserves another. This is the BABB after all. /phpBB/images/smiles/icon_smile.gif

Hale_Bopp

2002-Feb-04, 01:23 PM

Yes, you can always nitpick. But even the nitpicks have nitpicks and so on, until you get to the point where you are doing an impossible problem trying to account for every little thing (the car is moving, so its contribution to the gravitational field now has a time dirivative!)

Me, I say we all just assume spherical chickens, if you catch my drift /phpBB/images/smiles/icon_smile.gif

Rob

kilopi

2003-Jul-08, 01:54 AM

even the nitpicks have nitpicks and so on, until you get to the point where you are doing an impossible problem trying to account for every little thing

Yahbut, in this case, it's helpful.

For instance, I just opened to page 281 in Fundamentals of Astrodynamics by Bate, Mueller and White, and I thought of this thread. It's a figure showing the Geometry of the ballistic missile trajectory, and of course it's an ellipse. The ellipse intersects the Earth, but it is an ellipse. Even a thrown ball could be considered to be in orbit--and it would be, if it weren't for atmospheric and lithospheric friction.

Kaptain K

2003-Jul-08, 06:51 AM

even the nitpicks have nitpicks and so on, until you get to the point where you are doing an impossible problem trying to account for every little thing

Yahbut, in this case, it's helpful.

For instance, I just opened to page 281 in Fundamentals of Astrodynamics by Bate, Mueller and White, and I thought of this thread. It's a figure showing the Geometry of the ballistic missile trajectory, and of course it's an ellipse. The ellipse intersects the Earth, but it is an ellipse. Even a thrown ball could be considered to be in orbit--and it would be, if it weren't for atmospheric and lithospheric friction.

Hmmmm I always thought a ballistic trajectory was a parabola. Of course, a parabola is just an ellipse with an infinite major axis! :wink:

kilopi

2003-Jul-08, 10:52 AM

Hmmmm I always thought a ballistic trajectory was a parabola. Of course, a parabola is just an ellipse with an infinite major axis!

Yep, exactly why it's not a parabola. In order to be a parabola, it'd have to have escape velocity--or as David Simmons points out, the gravity field would have to be constant and uniform. Constant and uniform gravity is a good approximation for short distances and small velocities, like a thrown ball.

That thrown ball would go into orbit, if it didn't keep running into the Earth--that is, if the Earth had the same amount of mass but a lot smaller radius.

[fixed formatting]

Kaptain K

2003-Jul-08, 02:53 PM

I think I see your point. If the Earth were a point mass and I were to stand on a platform (held up by a rocket motor) 6400 Km above, a ball thrown any direction (except vertically) would trace an ellipse and return approximately 90 minutes later. Thanks for the insight. 8)

kilopi

2003-Jul-08, 03:05 PM

Thanks for the insight.

We both should thank David. It just took me a while to realize that a ball thrown up in the air is actually "in orbit," in a sense. The so-called orbital velocity for a particular radius is only for a circular orbit.

It all fits. :)

ToSeek

2003-Jul-08, 04:20 PM

Hmmmm I always thought a ballistic trajectory was a parabola. Of course, a parabola is just an ellipse with an infinite major axis!

Yep, exactly why it's not a parabola. In order to be a parabola, it'd have to have escape velocity--or as David Simmons points out, the gravity field would have to be constant and uniform. Constant and uniform gravity is a good approximation for short distances and small velocities, like a thrown ball.

If the Earth were flat rather than round, would a ballistic trajectory be parabolic rather than elliptical? Or is that basically what you're saying with regard to "constant and uniform gravity?"

Glom

2003-Jul-08, 04:32 PM

A parabola is the approximate shape when dealing with height ranges such that gravitational field strength can be regarded as constant. It's only elliptical when you consider the inverse square law. It's simply a case of suitable approximations.

If Earth were flat, the G-field would be screwy and would probably indeed result in a parabola. A zero dimensional mass (a point mass) has a G-field following inverse square law. A one dimensional mass (an infinite rod) has a G-field following the linear inverse law. Therefore, I'd imagine that a two dimensional mass (a plane such as a flat Earth) would have a constant G-field. It's probably like the field between two charged plates. Hence, the ballistic trajectory would be parabolic.

daver

2003-Jul-08, 06:12 PM

Therefore, I'd imagine that a two dimensional mass (a plane such as a flat Earth) would have a constant G-field. It's probably like the field between two charged plates. Hence, the ballistic trajectory would be parabolic.

Yes. Larry Niven in one of his essays described a platter world--imagine a cd the size of the solar system, with the sun bobbing up and down through the hole in the middle. The sun never gets very high above the horizon; Niven thought it would be a great place for gothic horror stories. Anyway, as long as you stay away from the edges, the gravitational field would be nearly constant for all reasonable trajectories (which means you're going to have to be very careful to ward off comets--when they come in, they'll come in FAST). So baseballs, cannon balls, missiles, would follow basically parabolic trajectories.

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