Consider a head-on collision between two particles of masses m1 and m2.The initial speeds of the particles are u1 and u2 in the same direction.The collision starts at t=0 and the particles interact for a time interval △t.During the collision,the speed of the first particle varies as
v(t)=u1+t△t(v1−u1).
Using law of consetrvation of momentum m1u1+m2u2=m1v(t)+m2v1
where v=speed of 2nd particle during collision.
⇒m1u1+m2u2
=m1u1+m1.(t△t)(v1−u1)+m2v1
⇒m2u2m2=m1m2(t△t)(v1−u1)+v1
∴ v1=u2−m1m2t△t(v1−u1)