Orbital mechanics? can someone explain to me how orbits work?

Question

I'm confused by the canon analogy where a cannon on a mountain shoots and the ball falls at the rate as the curve of the earth falls beneath it. But wait a minute! things don't fall at a constant rate, they accelerate! so wouldn't the curve have to 'accerlate' also (some weird spiral I guess)? I need a new persepective to understand orbits. equations are welcome

Answer 1

Note:  I did not do any of the thumbs down.

Your perception of the spiral is very insightful.

"the projectile will fall at the same rate (accelerate toward the earth) as the earth curves away from the projectile." is correct and:

The projectile requires energy (thrust) to maintain orbit, since gravity continues to act on it.  Without the extra thrust, the projectile will, as you say, spiral back to earth.

Satellites now in orbit, have thrusters the keep them in space.  Satellite controllers are people who monitor the position of satellites and correct any drift by firing the thrusters.

When a satellite nears it's end of life (ability to be controlled), it is supposed to use the last of it's fuel to push itself out into space.  Thus freeing that orbital location.

Answer 2

Knowing Kepler's laws will help.

1. The orbit of every planet (satellite) is on an ellipse with the sun (planet) at the foci.

2.A line joining a planet and the sun sweeps out equal areas.

3. The squares of the orbital periods of planets are directly proportional to the cubes of the semi-major axis of the orbits.

The satellite slows as it reaches apogee(fartherest point) and speeds up as it reaches perigee ( closest point)

Answer 3

Saying the ball falls at a constant rate means the acceleration is constant.  At any instant in time, the earth is traveling through space, roughly tangent to the sun.  With no force acting on it, it would just keep on going in a straight line, getting farther from the sun.  But the gravitational force from the sun accelerates it toward the sun.  Too weak, and earth still flies away from the sun.  A certain precise value, and it forms a circular orbit.  Near that, an elliptical orbit.  Too strong, and it will collide with the sun at the perigee of the orbit.  The reason gravitational orbits (and electric orbits) are stable, is the forces are exactly inverse square with distance.

Answer 4

As you increase the horizontal velocity of the projectile, eventually you reach a speed where the projectile will fall at the same rate (accelerate toward the earth) as the earth curves away from the projectile. The velocity of a projectile that becomes a satellite in circular orbit around the Earth depends upon the radius of the orbit and the acceleration of gravity at the orbit.

The orbital speed is

v(orbital) = sqrt(r g)   where g is measured at the orbal distance r