I have come over this problem where you are standing on the ice, whit a rope tied around your waist, which goes trough a pulley attached to a tree, and back to you again, as illustrated.
The question is, when you apply a force $F$, what is your acceleration. The answer options are $F/m$, $2F/m$, $3F/m$, and $4F/m$. My friends insist on the second option, due to the fact that you are pulling yourself with a force $F$, as well as the fact the rope pulls you with the same force, ergo $\Sigma F = 2F$. I find this incredibly counter intuitive. My way of attacking the problem is by first imagining two people standing on the ice with a rope between them, and one of the pulling with a force $F$. Due to the sum of external forces are zero, their combined center of mass will stay still, but they will both accelerate $F/2m$ relative to the ice. We then go to the same scenario, but the rope goes trough a pulley. I don't have any rigorous math underpinning this, but i believe they will still accelerate the same amount, but now also in the same direction. This seems to like the same scenario as we started with, just with double mass. My conclusion is therefore the acceleration is $F/m$. Is this correct? is the reasoning sound? Whats the best way to attack this problem?