Collision between two or more particles is the interaction for a short interval of time in which they apply relatively strong forces on each other.
In a collision physical contact of two bodies is not necessary. there are two types of collisions:
1. Elastic collision
The collision in which both the momentum and the kinetic energy of the system remains conserved are called elastic collisions.
In an elastic collision all the involved forces are conservative forces.
Total energy remains conserved.
2. Inelastic collision
The collision in which only the momentum remains conserved but kinetic energy does not remain conserved are called inelastic collisions.
In an inelastic collision some or all the involved forces are non-conservative forces.
Total energy of the system remains conserved.
If after the collision two bodies stick to each other, then the collision is said to be perfectly inelastic.
Coefficient of Restitution or Resilience
The ratio of relative velocity of separation after collision to the velocity of approach before collision is called coefficient of restitution resilience.
It is represented by e and it depends upon the material of the collidingI bodies.
For a perfectly elastic collision, e = 1
For a perfectly inelastic collision, e = 0
For all other collisions, 0 < e < 1
One-Dimensional or Head-on Collision
If the initial and final velocities of colliding bodies lie along the same line, then the collision is called one dimensional or head-on collision.
Inelastic One Dimensional Collision
Applying Newton’s experimental law, we have
Velocities after collision
v1 = (m1 – m2) u1 + 2m2u2 / (m1 + m2)
and v2 = (m2 – m1) u2 + 2m1u1 / (m1 + m2)
When masses of two colliding bodies are equal, then after the collision, the bodies exchange their velocities.
v1 = u2 and v2 = u1
If the second body of same mass (m1 = m2) is at rest, then after collision first body comes to rest and second body starts moving with the initial velocity of the first body.
v1 = 0 and v2 = u1
If a light body of mass m1 collides with a very heavy body of mass m2 at rest, then after a collision.
v1 = – u1 and v2 = 0
It means the light body will be rebound with its own velocity and heavy body will continue to be at rest.
If a very heavy body of mass m1 collides with a light body of mass m2(m1 > > m21) at rest, then after collision
v1 = u1 and v2 = 2u1
In Inelastic One Dimensional Collision
Loss of kinetic energy
ΔE = m1m2 / 2(m1 + m2) (u1 – u2)2 (1 – e2)
In Perfectly Inelastic One Dimensional Collision
Velocity of separation after collision = 0.
Loss of kinetic energy = m1m2 (u1 – u2)2 / 2(m1 + m2)
If a body is dropped from a height ho and it strikes the ground with velocity vo and after inelastic collision, it rebounds with velocity v1 and rises to a height h1, then
If after n collisions with the ground, the body rebounds with a velocity vn and rises to a height hn then
en = vn / vo = √hn / ho
Two-Dimensional or Oblique Collision
If the initial and final velocities of colliding bodies do not lie along the same line, then the collision is called two dimensional or oblique Collision.
In horizontal direction,
m1u1 cos α1 + m2u2 cos α2= m1v1 cos β1 + m2v2 cos β2
In vertical direction.
m1u1 sin α1 – m2u2 sin α2 = m1u1 sin β1 – m2u2 sin β2
If m1 = m2 and α1 + α2 = 90°
then β1 + β2 = 90°
If a particle A of mass m1 moving along z-axis with a speed u makes an elastic collision with another stationary body B of mass m2
From conservation law of momentum
m1u = m1v1 cos α + m2v2 cos β
sin β2v2 sin α – m1v1O = m