How will we derive at the expression for Newton's second law of motion?

Newton's second law of motion states that the rate of change of momentum of a body is directly proportional to the force and takes place in the same direction as the force. Suppose the velocity of a body of mass m changes from u to v in time t. The magnitude of initial and final momentum of the body will be
p₁ = mu ----(1)
and p₂ = mv ----(2)
The change in momentum (p₂ – p₁) takes place in time t. According to Newton's second law of motion, the magnitude of the force F is
F ∝
F = k ----(3)
where k is a constant of proportionality. But substituting equation (1) and (2) in (3) we get
F = k
Now is the magnitude of the rate of change of velocity \\, which is the acceleration a, so we have
F = kma ----(4)
We can choose the units of force in such a manner that the value of k = 1. If we substitute this value in equation (4), we getF = ma
The above equation states that the product of its mass and the acceleration determines the force acting on a body.