Everybody continues in the state of rest or uniform motion unless an external force is applied to it. 

it gives a qualitative analysis of force because of the following reasons. 

1. It shows the state and configuration of the body. 

To prove it:

We have,

F =  dpdt   (from the second law of motion)

If no external force is applied.

F = 0

Then,

dp/dt = 0

The above equation shows that momentum is constant which finally illustrates that the body had not changed its state or configuration.

 Newton's Second law of motion 

The rate of change of linear momentum with respect to time is directly proportional to the net external force applied.

F  ∞  dpdt

F   = k dpdt     k=1, this value of k is experimentally found and is   

                          Universally constant for all body)

We know,

P=m.v

F  =  d(m.v)dt       since mass is constant

    = m dvdt

Newton's third law of motion 

To every action, there is an equal and opposite reaction.

To prove this:

Fig: Two body collide with each other

We find,

And also,

From equation 1 and 2, we can finally explain force act in opposite direction and also do not cancel with each other because they act on a different body. These forces are alike in magnitude but opposite in direction. 

                                        Impulse

It is the average force act in a small interval of time. Mathematically, we find impulsive to be equal with the rate of change in momentum of the body.

And also,

It can be written as:

There is a wide application area where the concept of impulse is employed.

1. In the field of transportation of weak and sensitive body like glasses china clay and many others.

Glasses and china clay are packed and wrapped in thicker bags because of their sensitivity. During their delivery and transportation, they collide with each other and get broken. So as to avoid this it is packed in a thicker bag.

                        To prove it mathematically

             Mass pulley system

Let “T” be the tension act in the rope and rope be frictionless. Here M1 and M2 is the mass of two body held in two corners of the rope.

Here,

M2 > M1

So that system will accelerate toward a bigger mass. Let a be the acceleration produced in the rope.

We know,

                                     For mass M1

Tension on the rope is higher than the weight produced by mass. So the system will automatically move in the direction of heavy mass.

                                   For mass M2

Since the mass of M2 is higher than M1. Acceleration of the system act toward the heavier mass direction. In this case, the body moves the downward weight of the body must exceed the tension on the rope.

On solving equation 1 and 2, we find

                    The apparent weight of the body

1. When the lift is at rest

2. When lift is moving up and down.