**Elastic collision**

An elastic collision is that in which the momentum of the system as well as kinetic energy of the system before and after collision is conserved.

**Inelastic collision**

An inelastic collision is that in which the momentum of the system before and after collision is conserved but the kinetic energy before and after collision is not conserved.

**Elastic collision in one dimension**

Consider two non-rotating spheres of mass m1 and m2 moving initially along the line joining their centers with velocities u1 and u2 in the same direction. Let u1 is greater than u2. They collide with one another and after having an elastic collision start moving with velocities v1 and v2 in the same directions on the same line.

Momentum of the system before collision = m1u1 + m2u2

Momentum of the system after collision = m1v1 + m2v2

According to the law of conservation of momentum:

m1u1 + m2u2 = m1v1 + m2v2

m1v1 – m1u1 = m2u2 – m2v2

m1(v1 – u1) = m2(u2 – v2) ——-(1)

Similarly

K.E of the system before collision = ½ m1u12 + ½ m2u22

K.E of the system after collision = ½ m1v12 + ½ m2v22

Since the collision is elastic, so the K.E of the system before and after collision is conserved .

Thus

½ m1v12 + ½ m2v22 = ½ m1u12 + ½ m2u22

½ (m1v12 + m2v22) = ½ (m1u12 + ½ m2u22

m1v12-m1u12=m2u22-m2v22

m1(v12-u12) = m2(u22-v22)

m1(v1+u1) (v1-u1) = m2(u2+v2) (u2-v2) ——- (2)

Dividing equation (2) by equation (1)

V1+U1 = U2+V2

From the above equation

V1=U2 +V2 -U1_________(a)

V2=V1+U1 -U2_________(b)

Putting the value of V2 in equation (1)

m1 (v1-u1) =m2 (u2-v2)

m1 (v1-u1) =m2{u2-(v1+u1-u2)}

m1(v1-u1)=m2{u2-v1-u1+u2}

m1(v1-u1)=m2{2u2-v1-u1}

m1v1-m1u1=2m2u2-m2v1-m2u1

m1v1+m2v1=m1u1-m2u1+2m2u2

v1(m1+m2)=(m1-m2)u1-2m2u2

In order to obtain V2 putting the value of V1 from equation (a) in equation (i)

m1 (v1-u1) = m2(u2-v2)

m1(u2+v2-u1-u1)=m2(u2-v2)

m1(u2+v2-2u1)=m2(u2-v2)

m1u2+m1v2-2m1u1=m2u2-m2v2

m1v2+m2v2=2m1u1+m2u2-m1u2

v2(m1+m2)=2m1u1+(m2-m1)u2

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