# Conservation of linear momentum for a particular situation

Imagine a situation in which a man is standing on a floor. The floor is free from any external force (you can imagine it like the floor is resting on another friction-less floor), though gravity acts on the man and there can be friction between the man and the floor. If the man starts running in one direction, the floor starts moving in the opposite direction to conserve the linear momentum.

But to a fixed observer outside the "floor and man" system, the entire system moves as the floor (with the man on it) moves in the direction opposite to the direction of running man. If the observer cannot see the running man (suppose if everything above the floor is curtained) then it appears as if the system gained momentum without any external force acting on it. Isn't this a violation of the law of conservation of linear momentum? Can anybody explain the situation, please?

• The man is still part of the system, even if he's invisible to the observer. He's still there putting work into the system, so no violations occur. – hebetudinous Aug 2 '16 at 17:05
• yes, the man is part of the system but wouldn't any force exerted by him be considered as an internal force ? The law of conservation of linear momentum states that if no net external force acts on a system, it's linear momentum remains constant. pls explain – D.K. Aug 2 '16 at 17:10
• The total linear momentum remains constant; Add the man's linear momentum relative to the floor to the floor's linear momentum relative to the man and they'll sum to $0$. – hebetudinous Aug 2 '16 at 17:11
• The floor and the man will have opposite, but same in magnitude, velocities with respect to each other. Since, the mass of the man may not be equal to mass of the floor, therefore , their linear momentums wrt each other may not cancel up. Though i think it will cancel up if the velocities are taken with respect to the fixed observer. – D.K. Aug 2 '16 at 17:16
• Not the same velocities. Definitely not. The man will move faster along the floor than the floor moves along the frictionless floor beneath it. – hebetudinous Aug 2 '16 at 17:19

## 1 Answer

To avoid confusion I shall alter the problem so that the man is standing on a plank which itself rests on a frictionless floor.

The man and plank comprise an isolated system as far as horizontal forces are concerned. Any force (eg friction) between the man and plank are internal forces. If the man walks/runs in one direction, the plank moves in the other direction. Momentum is conserved, and as a result the centre of mass of this system remains in the same place.

Just because the man somehow becomes invisible to the observer does not mean that there is a violation of the Conservation of Momentum. You just have to look harder to find the missing momentum. The same problem of apparently missing momentum led to the postulation of the neutrino in 1930 by Wolfgang Pauli.

The man and the plank have equal and opposite momenta at all times, but not equal and opposite velocities. The heavier plank moves more slowly.

• Yes, that's the conclusion i came to at the end. thanks – D.K. Aug 3 '16 at 6:56