Consider a head on collision between two bodies of different masses. As seen in the figure if F(t) is the force acting on one body during the collision then -F(t) is the force acting on the other body. These forces will change the linear momentum of both the bodies.

This change depends on two factors and there are:

1) The average values of the forces

2) The time during which they are in contact.

According to Newton Second law, this can be represented as,

dp/dt = F(t) (for the body on which +F(t) is acting).

This can be written as ,

dp = F(t) dt

Here F(t) is the time varying force, as shown in the graph below

Now integrating both the sides over the collision time dt. If we assume that at time ti the linear momentum is pi and at time tf the momentum is pf , we have

So the result of the integration is

The right hand side in the above equation is called IMPULSE (J) of the collision.

Mathematically, Impluse is equal to the area under the curve F(t).

So, the change in the linear momentum of each body in a collision is equal to the impulse that acts on the body.

Pf - Pi = dP = J.