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According to (Force)*(Time) = (Impulse), if 50 Newton is exerted on the 2kg mass through a string in 0.5 seconds, then the hanging mass's change of momentum would be 25kgm/s.
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We can't say that. You're missing the other forces in the picture. If a body is subjected to a
constant net force of F newton for t seconds, its change in momentum will be, magnitude of F multiplied by t, yes. Here the 50 N isn't the net force.
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.....if in an experiment one pulls on the string with 50N tension and the string takes 0.5 seconds to completely transfer that force to the 2kg hanging mass...
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Force isn't a substance to be transferred. It is applied.
And regarding the situation you mentioned where the string breaks, if the string had indeed maintained a constant force of 50 N for the time 0.5 s on the body under the influence of
no other forces before breaking, then yes. the body should have a change in momentum of 25 Kgm/s.