Gravity is not like a string, or even a rubber band. It's a force that is proportional to the product of the masses and inversely proportional to the distance between the masses.
So if you are orbiting the Earth and you fire your engine to thrust you in the direction you are travelling (that is, tangential to the orbit) you get into a larger orbit. And the farther out you get from Earth, the less the pull from the Earth is. If you time it right, and head toward the Moon, the pull from the Moon gradually gets stronger as the pull from Earth gets weaker. Eventually the pull from the Moon on the spacecraft is stronger, and if you calculated correctly so that you're going in just the right direction at just the right velocity, you enter lunar orbit.
You see; this is where it gets all silly in their explanation for all this stuff.
First of all, we are told that a swingball is a good analogy for centripetal force in terms of showing us an orbit.
We could actually equate it to a hammer thrower at the olympics, the hammer being the craft and the chain being the gravitational force pulling on that ball, with the person being the Earth. The person let's go of the chain and the ball flies in a straight line out into the field.
So let's equate all that to the Apollo craft.
The craft enters space and it immediately orbiting the Earth at whatever speed. It's now in motion like the hammer thrower spinning his ball and chain.
Ok, now the Apollo craft has to sling shot towards the moon at the right time, so now it has to release itself from it's fast orbit around the Earth, just like the hammer thrower has to release the ball and chain.
The only way I can think of this happening, is for the craft to arrest it's orbit and channel that into a deflection of that velocity, or slingshot itself.
There is no possible way I can see how it could do this in a vacuum for starters.
The only possible way for it to happen in fantasy world would be for the craft to be held on by some invisible anchor that releases its shackle and channels that orbiting velocity into a straight velocity.
If I'm wrong, then explain in simple terms how I'm wrong. Try not to use magic and if possible, use an Earth analogy for what is happening, then equate it to what happens in space.
Convince me.
So there's never any "cutting of the string". I can't really think of a good analogy here to what's happening. It's not like a spring or rubber band, because the force on a spring gets bigger the more you stretch it. It's more like taffy, that you stretch out and it's hard at first, then it weakens. But then taffy doesn't really pull back either. Really the best thing to do is just use an actual example like the Newton's cannonball.
Newtons cannon ball makes no sense at all. If you can show me an Earthly movement that verifies this, I'll seriously look at it.