I’m confused though I am easily confused. So if she can generate some thrust towards the spaceship no matter how slow her movement is towards the ship she will make it there if the ship does not accelerate?
So even if she is moving at 1 foot an hour after giving one final burst of thrust she will eventually some day make it to the ship?
I would assume that the thrust she created would have to propel her at least a tiny bit faster than the ship was moving or she would never catch it.
You appear to be shifting your frame of reference in these two sentences.
The 1 ft per hour velocity only makes sense if it is relative to the ship - and she is moving toward it. So in some other frame of reference where the ship is moving, she will have a higher velocity than the ship.
We’re talking about the joys of “relative motion”. The point of interest is the difference in velocity between the two of them.
Let’s assume we put the space ship and astronaut in the middle of nowhere*. They can be sitting dead still or both traveling at 500 km per hour, as long as they are moving together, there is no difference for the purpose of our analysis. Since there’s no atmosphere, and thus no friction, they are effectively at rest.
Now our astronaut needs to get to the space ship. If she imparts any velocity towards the space ship, because there is no friction, there is nothing to slow her down. Thus she will get to the space ship eventually.** The more thrust, the faster she gets there.
So yes, she has to be propelled a tiny bit faster than the ship, but then the ship is no longer running away from her as fast as she is chasing after it.
The problem for this movie is they are not in the middle of nowhere - they are in orbit around Mars. Orbit is not a straight line path, it is a curved path***. The gravity of Mars makes the paths curve. That means that even though they start out on the same trajectory, if they differ very much in location, that difference in location will become a different orbit, which means their paths will not coincide, they will move apart.****
That movie situation is even more complicated by the fact that the astronauts originally arrive at some location in orbit along a path, then start looking around for the space craft. They see it below them, traveling in a somewhat different trajectory. It looks to me that the space craft is actually outrunning them. Erg. If they don’t act fast, they will never get there, because the ship will outrun them.
So astronaut 1 changes his vector and tries to catch the space ship. He says he’s going for an “overshoot”, but he doesn’t actually accomplish an overshoot, he does a direct intercept. Technical detail mangled for the movie, not the first or last in that movie.
So he thrusts and chases after the vehicle from behind and above, manages to bump into the vehicle and fasten the grapple from the tether, but cannot grab on himself. So he is receding from the vehicle in a somewhat down and forward vector from the vehicle path.
Now astronuats 2 through 4 arrive using the tether to pull themselves in. They sit on the vehicle and argue over the plan, all the while watching him drift further away. Finally the female astronaut goes on her rescue attempt. She uses her thrusters to start moving on a similar trajectory as astronaut 1.
Given the time scale all of this plays out, I can see that their trajectories do not drift that much. This all happens in 5 minutes. However, there will be some drift associated with the different trajectories.
Couple this with how the scene plays out. She thrusts to some halfway point and then stops her pursuit. Then she fires the tether. It comes up just short. Except if they were really moving at the speed difference from the vehicle as the movie portrayed when astronaut 1 collided with it, he’s already receding away from that point 2 feet beyond the end of the tether. They did not display a significant relative velocity between her and him. So either she matched his speed, in which case both of them are still getting further away from the vehicle, and thus that 50% fuel thing is all wrong. Or else she just missed catching him, and if she hadn’t hesitated so long before starting, she would have made it, and the time it took to recoil the tether for the second shot would have seen the distance between the two of them grow so far that he would have started to shrink in perspective size. Like a car driving away from you at 30 mph. It’s not going to sit “just out of reach” for long.
*If they are in the middle of nowhere, then what reference would we use to determine they are moving at 500 km per hour? Answer: we assume they are somewhere in space distant from any particular object, but still within the solar system. Thus the gravity from any objects is negligible but they are still close enough to some other object (i.e. the Sun) to use it as a reference for their velocity. Good enough?
**Eventually meaning “after the heat death of the universe” if necessary. The beauty of no other objects around to affect their paths. No friction or drag.
***Orbits are ellipses, sometimes eccentric ellipses, sometimes non eccentric ellipses. We call the later “circles”.
****If both objects are orbiting around the equator of the object, then their velocity vectors are pointed exactly into the orbital rotation. Then the drift will only be assocated with distance from the planet and forward or backward in the orbital path with respect to each other. However, most orbits have some inclination from the equator. This means the orbits trace a spiral path around the spherical planet. The distance between them will lead to tracing different routes over the surface of the planet. And of course I’m not talking about geostationary orbits.
Maybe they were in the Eve Online universe.