# Elastic and Inelastic Collision

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• When two bodies moving along a straight line collide with each other the collision is called the head-on collision. In a head-on collision, the initial and the final velocities are along the same straight line.

### Elastic Collision:

• The collision in which the total kinetic energy, as well as total momentum, is conserved is called elastic collision.
• In perfectly elastic collision the relative velocity of approach before collision is equal to the relative velocity of separation after collision.

#### Proof:

• Let us consider two bodies having masses m1 and m2  moving in the same direction, along the same straight line with velocities u1 and u2 respectively ( u1  >  u2). Let v1 and v2 be their velocities after the collision. Since all the velocities are in the same direction, we can write equation of conservation of momentum in scalar form.

m1u1+ m2u2 = m1v1+ m2v2

∴  m1u– m1v  = m2v2–  m2u2

∴  m1(u– v1)  = m2(v2–  u2)  ……….. (1)

For elastic collision, total kinetic energy before collision is equal to total collision after collision.

• The quantity u1 – u2 is called relative velocity of approach and the quantity v– v1 is called velocity of separation. Thus for elastic collision The relative velocity of approach before collision is equal to relative velocity of separation  after collision.

• We have,

#### Case – I

• When two spheres have equal masses i.e. m1 = m2 =  m

Similarly

Thus  in this case the two bodies exchange their velocities during collision.

#### Case – II

• The sphere B is at rest i.e. u2 = 0

Further if m1  = m2  = m,

∴ v1  =  0    and v2  = u1

Thus the body of mass m1 is stopped cold and body of mass m2 takes off with velocity which is equal to the initial velocity of body of mass m1 before collision.

### Inelastic Collision:

• The collision in which the total momentum is conserved but the total kinetic energy is not conserved is called the inelastic collision.
• A collision between two bodies is said to be a perfectly inelastic collision if they stick to each other and moves together with common velocity after collision.
• Let us consider two bodies having masses m1 and m2 moving in the same direction along the same straight line with velocities u1 and u2 respectively. ( u1 >  u2)

• Let v be the common velocity of the two bodies after the collision. Since all the velocities are in the same direction, we can write the equation of conservation of momentum in scalar form.

m1u1 +   m2u2 =   m1v  +  m2v

m1u1  +   m2u2 =   ( m1 +  m2)v

• Since the quantities on right sides are positive. The initial kinetic energy is always greater than the final kinetic energy in a perfectly inelastic collision.
• When two bodies collide with each other part of its kinetic energy gets converted into sound energy, heat energy etc. Therefore, total kinetic energy after collision decreases due to loss of K.E.

The Coefficient of Restitution:

• The ratio of the relative velocity of separation to the relative velocity of approach, in a collision between two bodies, is called the coefficient of restitution. It is denoted by u.
• For perfectly elastic collision e = 1 , For perfectly inelastic collision e = 0, for all other collisions 0  <  e   <  1

Action and reaction are two forces equal in magnitude and opposite in direction. Why they do not cancel each other?

When two forces having the same magnitude and opposite direction act on the same body their effect is canceled. Action and reaction forces are equal in magnitude and opposite in direction but they act on two different bodies. Therefore they do not cancel each other.

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