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Derivation of Newton’s Second Law of Motion

Newton’s second law of motion states that the rate of change of momentum of an object is Proportional to the applied unbalanced force in the direction of force.

Suppose an object of mass, m is moving along a straight line with an initial velocity, u. It is uniformly accelerated to velocity, v in time, t by the application of a constant force, F throughout the time t. The initial and final momentum of the object will be, p1 = mu and p2 = mv respectively.
The change in momentum =  p2 – p1
The change in momentum = mv – mu
The change in momentum = m × (v – u).
The rate of change of momentum = m × (v – u)/t(v - u)t
Or, the applied force, F ∝ m × (v – u)/t (v - u)t   
 F = km × (v – u)/t

F = kma     ---------------------------- (i)
Here, a  is the acceleration [i.e., a= (v – u)/t], which is the rate of change of velocity. The quantity, k is a constant of proportionality.

The SI units of mass and acceleration are kg and m s-2 respectively. The unit of force is so chosen that the value of the constant, k becomes one. For this, one unit of force is defined as the amount that produces an acceleration of 1 m s-2 in an object of 1 kg mass. That is,     
1 unit of force = k × (1 kg) × (1 m s-2).
Thus, the value of k becomes 1.

From Eq. (i)

F = ma 
The unit of force is kg m s-2 or newton, represented as N.