Can you used a tethered "boot" to move in space?

Question

Preface: On Earth, with the help of gravity, I can easily tie a wrench to a rope, then spin it around, lasso-style, and let go at any time to send the wrench and rope flying away from me in the air. I want to know if something similar is possible in space.

I know that movement through space is accomplished by delta-v, which typically means to get somewhere you must leave something behind. I'm just trying to figure out if that must always be the case.

Let's say I am only six feet away from my space ship when I realize that I forgot to attach the tether. The tether is perhaps twenty feet long, and is on a spool connected to my suit. I have no means to puncture my suit, and I am moving with exactly zero relative velocity to my space ship. Suppose I was using my MMU, and it just ran out of fuel right as I caught up to the space ship, but remained six feet from it. Now, I want to reach the space ship, which has plenty of grippy surfaces on it that I can use to get back inside as long as I can reach them. I also have a very massive wrench. Do I have any options?

Let's say I attach the wrench to the tether, spool out all 20 feet of the tether, but then grab it such that the wrench is attached about three feet from my hand. That leaves 17 feet of slack from my hand to the point where it is attached to my suit. Now, I want to shove (throw) the wrench toward the ship with all of my might. That's going to move me away from the ship by some amount, but then I want to try to start the wrench spinning in a lasso-style movement perpendicular to the ship (i.e. circle's radius would go from me out toward the ship). Assuming I can get the wrench moving lasso-style, I would let go of the tether just as the wrench was moving toward the ship, and then let it pull the cable and me along with it. When this happens in space, would I be flung in the opposite direction with equal force?

The idea I'm going for is that I can inject a lot of energy into the rotational movement of the wrench with my lasso technique. I'm burning calories doing this, so I'm not creating energy. I'm converting my stored energy into kinetic energy in the rotating lasso system. Is that even possible to do in space? Am I violating any laws of physics by trying to use angular momentum to induce a linear velocity?

I understand that if I throw the wrench, then pull the wrench, I will end up with a net movement of zero relative to my initial point. If the wrench has a sufficiently smaller mass to mine, then perhaps I could make the wrench move away from me fast enough to reach the space ship before I move more than the length of the tether away, and hope to hook something on the ship. Maybe the wrench could have a magnet on it. I don't know. Perhaps I have a gyroscope with me that I can hold onto for starting the lasso-maneuver.

Answer 1

The easiest way to think of this is probably to consider that since you have no external forces acting on you, your total center of mass can't move. No amount of calisthenics can change this. The best you can hope for is to throw the wrench straight at the spaceship. So the question is just, where will your body need to be when the wrench is at the spaceship, in order to keep the total COM stationary? That's simply a matter of mass ratios. If you and your suit weigh a hundred times what your wrench does, your effective reach will be 20' times $100/101$.

Answer 2

Spinning the wrench on a rope like a lasso is not very productive: It leaves you spinning in position, with rotation opposite to the wrench around you. On earth you can spin a lasso over your head, because you can apply torque with your feet against the ground. In space, you can only spin something around your center of mass. Very soon you will get entangled by the rope, and stop spinning.
Net change: you look like you encountered a hungry spider.

Throwing your wrench AT the ship, with or without a tether, is a form of suicide.
The wrench will move towards the ship, and you will move proportionally away from the ship.
If the tether runs out before the wrench hits the ship you will not have gained or lost anything, except time. Hauling in the wrench by pulling in the rope will move you back to your starting point.
Net change: just some effort and time wasted, no change in position.

Worst case: If the wrench HITS the ship, it will stop and possibly bounce back a bit. In the process imparting a bit of movement to the ship, and reducing the momentum you can recover from the wrench via the tether.
Net change: You and your wrench drift away from the ship at a slow but constant velocity.

There are two ways you can save yourself:
Your can either snag something on the ship, using a magnet on your wrench or entangling its tether around a part of the ship. Note that this had better work on the first try, as it is just a special case of hitting the ship with your wrench.
Net change: You will drift away, unless the snag with the ship can be used to pull you in.

Or the best possible scenario:
Untie your wrench from its tether. Bye-bye favorite tool.
Gently throw your wrench directly AWAY from the ship. It will drift off in the distance, hopefully never to be seen again.
Meanwhile, you will be drifting in the opposite direction with equal momentum. As you mass more than the wrench, you will drift slower. Slow is good, it gives more control. Eventually you will gently bump up to the ship. You now have one chance to grab on!. This is why you threw the wrench only gently, so that you will be moving slowly when you reach the ship, giving ample time to secure yourself.
Net change: You save yourself, and condemned Mr. Wrench to the depths of space.