Physically Incorrect

This is the solution to this puzzle.

1. As observed by this blog’s readers, the forces act on different bodies and hence cannot be added. The ground exerts a frictional force on the person, pushing him forward; the person exerts a force equal in magnitude and opposite in direction on the ground. Since the ground’s mass is “infinite” it doesn’t move (or moves by a tiny, insignificant amount).
2. Actually, it will! Let’s focus on a single molecule of air. First, the fan “throws it forward” with some momentum, mv. Consequently, the molecule pushes the boat back by the same amount, Δp1= -mv. Next, the air molecule collides with the sails. If this collision was inelastic, the molecule would stop upon impact and transfer its momentum, mv, to the boat, Δp2= +mv; in this case the total momentum transfered would be zero, Δp1+Δp2=0, and the boat wouldn’t move. However, while not elastic, the molecule-sails collision is not completely inelastic either, and the air molecule slightly bounces back upon hitting it. This means its change is momentum is greater than mv. For a perfectly elastic collision it would be 2mv, while in reality mv<Δp2<2mv and Δp1+Δp2>0, meaning the boat would propel forward. Although this works it’s extremely inefficient, and this is while neglecting air drag! A much better (although still inefficient) solution would be to mount the fan on the back of the boat facing away from the sails, where each molecule will fly off and transfer its momentum, mv, in its entirety to the boat.
   
3. The guy will be able to pull himself up. Just think about the guy + platform as a single unit (suppose his feet were glued). He pulls the rope and creates a tension T, so the rope pulls him up by the same force. This force gets transmitted along the rope’s length (because the rope is massless) and will also pull the platform up by the same tension. In fact, the rope’s end that’s connected to the platform will double the force the guy applies, although  the guy will need to support both his and the platform’s weight, so he’ll need more muscles to exert this larger force compared to, say, if he had been holding on to a rope by himself.

   

4. As in the previous question, a simple force diagram will help immensely (see below). You really shouldn’t care what’s inside the dashed frame.  the moment you’re told a rope is massless it will “transmit” the tension throughout its length. Now imagine the guy (having mass M) climbing up the rope by exerting some force F, so:  MA = -Mg+F (note that, in order for the guy to move up, F must be greater than Mg). The physical source of the force F is the rope’s tension,  which by Newton’s third law must equal F as well (only opposite in direction). So the tension T must equal F. The resulting force diagram for the block is MA = -Mg + F, precisely the same equation of motion governing the guy’s motion. They will therefore move by the same amount. This is also true whether the guy decides to climb down after reaching the top!