I asked, "When you 'place' your chunk of debris so that it will fall toward the planet, what stops your ship from falling along side of it?"

You answered,

Thrusters,... It doesn't really matter here, the important part is the debris suddenly falling, what happens to the ship doesn't matter as much.

What happens to the ship is crucial. You are proposing to power a generator on-board the ship by paying out a rope that is attached to the falling rock. The amount energy available for you to harvest, $W$ (for "work",) depends on the tension, $T$.

$$W=\int{}T(x)\,dx$$

If you keep the tension constant throughout the process, then that simplifies to $W=Tx$ where $x$ is the length of rope that your ship lets out.

The point is, if $T=0$ (no tension), then $W=0$ (no energy.)

If your ship is not rigidly attached to the planet (e.g., by being docked to a very tall tower that stands on the planet) then the only way I can imagine for your ship to maintain that tension is by continuous use of thrust.

I haven't taken the time to completely understand the situation, but I'll be surprised if the minimum amount of energy needed to work the thrusters turns out to be any less than the amount of energy you can harvest from the falling rock.

Actually, you said,

Thrusters, or simply flying the debris significantly closer...

It's not at all clear what you mean by that. Flying the debris closer than what? Your ship is transporting the "debris" from somewhere. The debris is travelling along with your ship. If your ship does not fire its thrusters, then how is it ever going to move away from the debris?

The ship must move away from the debris in order for you to harvest any energy. $W=Tx$. If $x=0$ (no movement,) then $W=0$ (no energy available for harvest.)

I asked, "When you 'place' your chunk of debris so that it will fall toward the planet, what stops your ship from falling along side of it?"

You answered,

Thrusters,... It doesn't really matter here, the important part is the debris suddenly falling, what happens to the ship doesn't matter as much.

What happens to the ship is crucial. You are proposing to power a generator on-board the ship by paying out a rope that is attached to the falling rock. The amount energy available for you to harvest, $W$ (for "work",) depends on the tension, $T$.

$$W=\int{}T(x)\,dx$$

If you keep the tension constant throughout the process, then that simplifies to $W=Tx$ where $x$ is the length of rope that your ship lets out.

The point is, if $T=0$ (no tension), then $W=0$ (no energy.)

If your ship is not rigidly attached to the planet (e.g., by being docked to a very tall tower that stands on the planet) then the only way I can imagine for your ship to maintain that tension is by continuous use of thrust.

Also, you're going to have to spend some energy in order to change the trajectory of your space rock so that it heads toward the planet. I haven't taken the time to completely understand the situation, but I'll be surprised if the minimum amount of energy needed to work the thrusters throughout the whole process is any less than the amount of energy you can harvest from the falling rock.

Actually, you said,

Thrusters, or simply flying the debris significantly closer...

It's not at all clear what you mean by that. Flying the debris closer than what? Your ship is transporting the "debris" from somewhere. The debris is travelling along with your ship. If your ship does not fire its thrusters, then how is it ever going to move away from the debris?

The ship must move away from the debris in order for you to harvest any energy. $W=Tx$. If $x=0$ (no movement,) then $W=0$ (no energy available for harvest.)

I asked, "When you 'place' your chunk of debris so that it will fall toward the planet, what stops your ship from falling along side of it?"

You answered,

Thrusters,... It doesn't really matter here, the important part is the debris suddenly falling, what happens to the ship doesn't matter as much.

What happens to the ship is crucial. You are proposing to power a generator on-board the ship by paying out a rope that is attached to the falling rock. The amount energy available for you to harvest, $W$ (for "work",) depends on the tension, $T$.

$$W=\int{}T(x)\,dx$$

If you keep the tension constant throughout the process, then that simplifies to $W=Tx$ where $x$ is the length of rope that your ship lets out.

The point is, if $T=0$ (no tension), then $W=0$ (no energy.)

If your ship is not rigidly attached to the planet (e.g., by being docked to a very tall tower that stands on the planet) then the only possible way for your ship to maintain that tension is by continuous use of thrust.

I haven't taken the time to completely understand the situation, but I'll be surprised if the minimum amount of energy needed to work the thrusters turns out to be any less than the amount of energy you can harvest from the falling rock.

Actually, you said,

Thrusters, or simply flying the debris significantly closer...

It's not at all clear what you mean by that. Flying the debris closer than what? Your ship is transporting the "debris" from somewhere. The debris is travelling along with your ship. If your ship does not fire its thrusters, then how is it ever going to move away from the debris?

The ship must move away from the debris in order for you to harvest any energy. $W=Tx$. If $x=0$ (no movement,) then $W=0$ (no energy available for harvest.)

I asked, "When you 'place' your chunk of debris so that it will fall toward the planet, what stops your ship from falling along side of it?"

You answered,

Thrusters,... It doesn't really matter here, the important part is the debris suddenly falling, what happens to the ship doesn't matter as much.

What happens to the ship is crucial. You are proposing to power a generator on-board the ship by paying out a rope that is attached to the falling rock. The amount energy available for you to harvest, $W$ (for "work",) depends on the tension, $T$.

$$W=\int{}T(x)\,dx$$

If you keep the tension constant throughout the process, then that simplifies to $W=Tx$ where $x$ is the length of rope that your ship lets out.

The point is, if $T=0$ (no tension), then $W=0$ (no energy.)

If your ship is not rigidly attached to the planet (e.g., by being docked to a very tall tower that stands on the planet) then the only way I can imagine for your ship to maintain that tension is by continuous use of thrust.

I haven't taken the time to completely understand the situation, but I'll be surprised if the minimum amount of energy needed to work the thrusters turns out to be any less than the amount of energy you can harvest from the falling rock.

Actually, you said,

Thrusters, or simply flying the debris significantly closer...

It's not at all clear what you mean by that. Flying the debris closer than what? Your ship is transporting the "debris" from somewhere. The debris is travelling along with your ship. If your ship does not fire its thrusters, then how is it ever going to move away from the debris?

The ship must move away from the debris in order for you to harvest any energy. $W=Tx$. If $x=0$ (no movement,) then $W=0$ (no energy available for harvest.)