NEWTON'S SECOND LAW OF MOTION

It is most unfortunate that a significant number of teachers of science put forward Newton's Second Law as being Force = Mass x Acceleration.

If an unbalanced force acts on an object its momentum will change with time. Newton's Second Law states that the unbalanced force is directly proportional to the rate of change of momentum with time and is in the same direction as the force.

Momentum  p = mv and is a vector quantity. Hence, from the above definition F = dp/dt = d(mv)/dt.
Upon expansion we get  F = mdv/dt + vdm/dt.   Now, dv/dt = a.

Hence, F = ma + vdm/dt. This is the correct mathematically presentation of Newton's Second Law.
If we assume that the inertial mass of the object does not change then dm/dt = 0 , and Newton's Second Law reduces to F = ma + 0 = ma.

Consider an object of mass m having an initial velocity of v'. Its momentum (p) is mv'. Suppose an unbalanced force  F acts on the object for t seconds and it's new velocity is v''. Hence, its new momentum is mv''. Note, I have made the assumption that its mass has not changed. Let the change in velocity be v. Then, v = v'' - v,'

From Newton's Second Law we get  F = (Change in momentum) / (Change in time).

Hence, F = (mv'' - mv')/t = m(v''-v')/t = m(v/t) = ma.