Shouldn't Conservation of Momentum Be Conserved in this Problem?

[ColdFusion85]

Shouldn't Conservation of Momentum Be Conserved in this Problem?

Alright, I am deeply confused on this problem. It goes like this:

A big sledge is standing at rest on a horizontal icy surface. A rail is placed in such a way that the sledge can move only along a straight line. It is assumed that the sledge moves without friction. At one end of the sledge n persons stand. They all have the same mass.
These n persons can now leave the sledge in two different ways:

(a) All n persons run to the front of the sledge and jump off simultaneously, all with velocity v relative to the sledge. This velocity is along the direction in which the sledge is constrained to move.
(b) One at a time, a person runs to the end of the sledge and jumps off with velocity v relative to the sledge while the remaining persons stand still. This process continues until they have all left the sledge.

Demonstrate which of the two methods imparts to the sledge the highest velocity relative to the ice.


I would think that both would result in the same velocity, but if I had to pick one answer, it would be (a), since the total mass of all n people, M would impart a larger velocity V on the sledge than one individual would at a time. The correct answer is (b) however! The only thing I can picture is that the center of mass would be more towards the back if each individual jumped off one at a time, and since the velocity would be directed in that direction (3rd law, equal and opposite forces), this would impart the higher velocity. But then wouldn't this be the same as if all people jumped off and imparted the same total velocity V on the sledge? It seems that Conservation of Momentum should say that both cases would produce the same velocity, if one takes into account only external forces.

Can ANYONE explain as to why the correct answer is case B?

 

[quantumdude]

Why not try to work it out for yourself, plugging in some numbers? Take 3 people with mass 75kg who run off at 10 m/s relative to the sled. You'll see that if they all run off together, each person has to work against the mass of all the other people. And if they run off one at a time, each successive person has to work against a smaller mass than the one before him, thus each successive person has a greater effect on the final momentum of the sled.

 

[ColdFusion85]

wow that becomes so obvious when one looks at it from that point of view. basically you solve MV=mv for V, where V = velocity of center of mass. So we have V=(mv)/M. if they all jump off together they impart a velocity of 3.33 m/s,where as if they go off one by one, each successive person imparts a higer velocity each time. thanks a lot tom!