Question

Derivation of 2nd law of motion by differentiation method.

Answer 1

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
Or, the applied force,
F ∝m × (v −u)/t
F  = km (v -  u)/t

F = kma 
Here a [a = (v – u)/t ] is the acceleration, 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-2respectively. 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. and 
F = ma which is the mathematical expression on the Newton's second law of motion.

Answer 2

answer is in the attachment