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Consider a body of mass m ,moving with a velocity u initially after a time t its velocity changes to v then change in momentum p = pf -pi = mv- mu
rate of change of momentum = mv-mu/t
according to the above statement , 
F is directly proportional to mv-mu/t
F is directly proportional to m(v-u)/t
F is directly proportinal to ma 
here k is proportionality constantand its value depends on the definition of unit force.here k=1then F=ma
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