Question

The path of a projectile follows some parabolic path... But a satellite follows some elliptical path around the earth. When does the parabola of the projectile become elliptic? Or why isn't the path of a projectile described by and ellipse?

Answer 1

The parabola is a consequence of a uniform force-field.

Actually, the force-field is directed towards the centre of the Earth, and slightly non uniform. But it is not necessary to take that into account for short-range shots, stopped by presence of Earth's surface.

That parabola is the approximation of the top part of a very eccentrical ellipse.
(e ≈ 1, though slightly less)

The calculated path with constant acceleration is the osculating parabola at the perigee of the ellipse, which is the real path.

If the Earth were a mass-point C and no atmosphere present, then your projectile would orbit C in a full elliptic path with C being a focus.

Answer 2

Satellites travel elliptical paths with the center of the Earth at one focus. Anything shot from the surface of the Earth, a baseball, say, or a cannonball, travels an elliptical path, but the ellipse soon intersects the surface of the Earth again (we often say it's a parabola, and it is to very high precision, but technically it's the outer end of a very long ellipse.) Ballistic missiles do the same thing except their ellipses intersect the surface of the Earth thousands of kilometers away. Nothing shot directly from the surface of the Earth can go into orbit; it will either fall back to Earth again or, if it's moving fast enough, escape completely.

Incidentally, if we could somehow magically let the object pass through the earth's interior, it would not travel in an ellipse. One of the cool things about gravity is that, for a spherical object, the gravity is the same as if all the mass were at a single point in the center. That's if you're outside the planet. If you're inside the planet, the mass above you has no gravitational effect. Only the mass between you and the center counts. If the earth were perfectly uniform, gravity would decrease linearly toward the center and would be zero at the center - all the mass of the earth around you would be pulling in all directions equally. In the real earth, because mass is concentrated in the core, gravity actually increases with depth and is a few per cent higher at the core boundary than on the surface.

However, on the real earth, if we throw something up, it follows an elliptical path until it intersects the surface again.

Objects stay in orbit because of a balance between inertia, that would cause them to keep moving in a straight line, and gravity, that would pull them down. Isaac Newton conceived of artificial satellites. He pointed out that a cannon on a high enough mountain and firing ever faster cannonballs could fire them to greater and greater distances. If fired with a great enough velocity, the curvature of the cannonball's path would be equal to that of the Earth and the cannonball would circle the Earth.

To get into orbit, you have to climb Newton's mountain first. Rockets are launched into orbit by launching them vertically to get them above the atmosphere, then accelerating them horizontally to reach orbital velocity. It takes 29,000 km/hour to do this in low Earth orbit. You get 1670 km/hour of this for free thanks to the Earth's rotation. That's why most satellites are launched eastward.

Answer 3

I guess that was more than enough.. :P

Answer 4

^ya it was :D

Answer 5

it is said to be a parabola because we take 'g' to be a constant at small flights.. but in case of satellite we cant consider a constant 'g' approximation.. thus an ellipse. here 'g' varies with 'h'
then maths says from its equation of trajectory that it will be a form of an ellipse .. while constant g situation will give an equation of x and y distances in a form of an parabola's equation.