**Newton's Second law of motion**: **Newton's Second law of motion** states that the 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.

**Newton’s Third law of motion: Newton's Third law of motion states that for** every action there is an equal and opposite reaction.

**Derivation of Newton's Second Law**

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,

p_{1} = mu and p_{2} = mv respectively.

The change in momentum ∝ p_{2} – p_{1}

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^{-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. and

F = ma which is the mathematical expression on the Newton's second law of motion.

newton's third law of motion from second law

Consider an isolated system of 2 bodies A & B.An isolated system means where there is no external force acting.Now let FAB be the force acting on B by A & FBA be the force acting on A by B.Now rate of change of momentum of A =dpA/dt and rate of change of momentum of B =dpB/dt

thus FAB=dpB/dt (i)......

FBA=dpA/dt (ii)........

Adding 1 and 2 we get FAB+FBA=dpB/dt+dpA/dt =d(pB+pA)/dt

But if no force is applied momentum will also be 0 becoz no velocity will be there.so rate of change of momentum will also be 0

thus d(pA+pB)/dt=0

Read more on Brainly.in - https://brainly.in/question/954432#readmore