CALCULATION OF OBJECTS MOVING IN GRAVITATIONAL FIELDS

In the graph to the left these energy levels in a gravitational field are shown. To interpret the curves, firstly consider the situation at zero distance from planet surface. An object is then rolling over it at 'orbital' speed Ekorb (around 8 km/sec on Earth), meaning it is weightless, but just doesn't lift from the ground.

The launch energy is then the same as the kinetic energy Ekorb of the rolling object, because its potential energy Epw is zero there.

Observe, that if we instead would have used this energy to shoot the objects straight up, radially away from the planet, it will reach a height of one planet radius, where all its initial kinetic energy has been converted to potential energy, comes to a halt there and starts falling back - it can never get into orbit (unless at zero height).

From this we can conclude:

1) Any object in orbit, however low it is, requires more launch energy than to shoot it straight up to a height of one planet radius, which is half of the escape energy from planet surface!

2) To let an object in orbit escape, its kinetic energy must be doubled (it's speed increased with a factor square root of 2 = 1.41). This speed will then become less and less, as the object moves away from the planet, to become zero on infinite distance.

Confusing may be that the real potential energy Ep and the kinetic energy Ekorb both approach zero on infinite distance - where has the launch energy gone? Even a certain Mr. Einstein might have had a problem to answer that question (science still doesn't know what "gravity" is), so who am I to do it?

My speculation though is that a gravitational field maintains its energy content and thus whatever gets out, must be put in first. In consequence, as the object moves farther and farther away from the planet, its mechanical energy is gradually absorbed by the planet's gravitational field, being stored there - it's like paying a ransom to get out of prison. This also complies with our everyday's experience. Whatever energy we get out of gravity, had to be put in there first - nothing falls down, if it hasn't been lifted up before.. There could thus be a "Law of Conservation of Gravity" (which then applies on the whole of the Universe, giving the Gravity Constant, as measured).

The object is launched from a barrel (á la Jules Verne) and when it leaves its mouth, it has its maximum speed Vi (kinetic energy Ek = Em), that becomes lower and lower, as it gains height (potential energy Ep), at constant total energy Em (mechanical energy).

If the launch speed is higher than 7900 m/s, the object will go into orbit, if lower, it will fall back to Earth (atmospheric air resistance neglected here)

The acceleration of the object during launch depends on the length of the barrel. The lesser,the longer the barrel is.

Say for example it is 10 km long, then the acceleration becomes: a = 4691 m/s², which is 479 G. 1G gives your weight on Earth, thus if the object would be a manned space capsule, the people in there would become a blood pool on its bottom. If we above fill in the values of M and Ro for the Moon, we get instead 130 G, which still means a "blood pool" and it is yet only in orbit on one planet radius distance. To escape from Earth and the Moon, the G-values become 638 and 174 respectively.

Hence, who wants to launch anything through a barrel from Earth, or even Moon surface? If you ever would see such a proposal (you will), you can now "smile" at it, the more so, the shorter the proposed barrel is (and if several 100 km's long, how to build and operate it?)

To launch the object in the same orbit with a rocket engine (same Vo and height) , thus starting at zero speed on planet surface, we get the folowing result:

When the rocket engine stops, the object's speed Vm is higher than the targeted orbital one Vo, but's it mechanical energy Em equals the one required for that orbit.

Thus it will continue to move away from the planet at constant mechanical energy Em, while its speed (kinetic energy Ek) gets lower and lower and its potential energy Ep increases, until it reaches orbital speed and height, where energy balance is obtained: the object is now in steady orbit.

In fact, after the rocket engine stopped working, the situation becomes equal to the one above, but now the object being shot-launched from a certain height y.Ro over planet surface. (the according "barrel" would have to be that long!)

We see again that a gravitational field is characterized by energy levels. If an object has less mechanical energy than required for its position, it will either fall down, or have weight when resting on a support. We and everything around us has weight, because the energy is too low not to have it. Thus any space-elevator idea, saying that centrifugal forces would keep the thing upright, "hanging" in space, is wrong - it will have weight, period!

How this all applies on various space projects and settlements on the Moon, you can read here, the first and afterfollowing chapters of a Science Vision book about settlements on the Moon.