Orbits & Projectiles: Why the Difference?

[mtanti]

Hello everyone,

Why is it that in projectiles we learn that a mass launched horizontally will always fall towards gravity regardless of its horizontal velocity but in orbiting satellites this does not happen?

Thanks!

 

[Janus]

They do, it's just that the Earth curves out from under them as they do. The thing with projectiles fired horizontally on the Earth is that, unless you fire them with a very high speed, the Earth's curvature doesn't become enough of a factor to worry about. Both fall due to gravity, its just that once you launch them at a high enough speed, the resulting trajectory no longer intersects the surface of the Earth.

 

[mtanti]

Not sure what the curvature of the Earth has to do with it because gravity will still pull the projectile towards the surface, but wouldn't that just result in a spiral towards the Earth then? Perhaps the answer would help me understand why comets don't just go straight into the sun as they approach it?

 

[Oberst Villa]

Why not try it yourself ?

http://spaceplace.nasa.gov/en/kids/orbits1.shtml

(apologies for the Kindergarten style, if you google for Newton, Cannon and Mountain you might find some applets that look a bit more sophisticated)

EDIT: like this one: http://www.waowen.screaming.net/revision/force&motion/ncananim.htm

2nd EDIT: and finally here the classical illustration: http://www.waowen.screaming.net/revision/force&motion/ncanon.htm

 

[Janus]

Okay, I'm going to assume that you've been taught that if you fire a projectile horizontally, it will always follow a parabolic path until it hits the ground. This actually isn't true, because it assumes that the Earth is flat and that the force of gravity always acts in a direction perpendicular to the flat surface.

The Earth isn't flat however, and the force of gravity always points to the center of the Earth. The result is that the projectile actually follows an elliptical path, with the center of the Earth at one focus of the ellipse. (Except for the cases where the projectile is fired at or above escape velocity, in which it will take a parabolic or hyberbolic tracjectory with the center of the Earth at the focus)

As the speed at which the projectile is fired increases, the eccentricity of the ellipse changes until it equals 0, at which point, the trajectory becomes a circle (which is a special case of ellipse.) and you have acheived a circular orbit. If the projectile is fird at even a higher speed, the trajectory becomes more elliptical course, but one where it climbs away from the Earth and then returns to its original point, then repeats. It enters an elliptical orbit with a perigee at the original launch point.

All objects following ballistic paths with respect to a central gravitational body wil follow these types of trajectories. Comets follow either ellipitic or hyperbolic trajectories. If the perihelion (closest approach to the Sun's center) is outside the Sun's surface, the comet will miss the Sun on the close approach and head back out into space. (Some comets do strike the Sun, as there trajectories intersect the surface).

 

[mtanti]

Oh, so every trajectory is an elliptical path, and the faster a projectile goes, the larger the ellipse until it is larger than the earth. Correct?

 

[mtanti]

Is it even possible for something to spiral down into a planet, as in goes round and round and lower and lower until it crashes?

 

[Tw15t3r]

That is not possible if we assume that there is no air resistance, for a spiral can only be formed if its "curvature" keeps increasing, i.e. a spiral has to bend more and more as it goes inward.

In order for this "curveness" to increase, the horizontal speed of a projectile would have to be lowered continously, hence the need for some kind of resistance like air resistance.

 

[Oberst Villa]

Oh, so every trajectory is an elliptical path, and the faster a projectile goes, the larger the ellipse until it is larger than the earth. Correct?

Nearly correct. Look at what Janus wrote in post #5, 2nd paragraph:

"(Except for the cases where the projectile is fired at or above escape velocity, in which it will take a parabolic or hyberbolic tracjectory with the center of the Earth at the focus)"

In this cases the projectile escapes into infinity...

 

[mtanti]

Oh ok, it's clear now. Thanks!