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To understand this concept it is necessary to understand Newton's Second Law
According to Newtons Second law applied force is equal to rate of change of momentum of a body.
Mathematically,
here, is rate of change of momentum with respect to time
It means If two eggs fall from same height,one on softer surface and other on hard surface that time the momentum of both eggs will remain equal at both the surfaces. But, impact time will be different. On hard surface egg will stop almost instantly so impact time will be small and hence the force on egg will be large therefore the egg will breakup.
On the other hand on the soft surface like a cotton, egg will not stop instantly but it will slow down for a few seconds and then stop due to which the time of impact will increase. Therefore the force on egg will be less and it won't break up.