Collisions
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 nonconservative 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
OneDimensional or Headon Collision
If the initial and final velocities of colliding bodies lie along the same line, then the collision is called one dimensional or headon collision.
Inelastic One Dimensional Collision
Applying Newton’s experimental law, we have
Velocities after collision
v_{1} = (m_{1} – m_{2}) u_{1} + 2m_{2}u_{2} / (m_{1} + m_{2})
and v_{2} = (m_{2} – m_{1}) u_{2} + 2m_{1}u_{1} / (m_{1} + m_{2})
When masses of two colliding bodies are equal, then after the collision, the bodies exchange their velocities.
v_{1} = u_{2} and v_{2} = u_{1}
If the second body of same mass (m_{1} = m_{2}) is at rest, then after collision first body comes to rest and second body starts moving with the initial velocity of the first body.
v_{1} = 0 and v_{2} = u_{1}
If a light body of mass m_{1} collides with a very heavy body of mass m_{2} at rest, then after a collision.
v_{1} = – u_{1} and v_{2} = 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 m_{1} collides with a light body of mass m_{2}(m_{1} > > m_{21}) at rest, then after collision
v_{1} = u_{1} and v_{2} = 2u_{1}
In Inelastic One Dimensional Collision
Loss of kinetic energy
ΔE = m_{1}m_{2} / 2(m_{1} + m_{2}) (u_{1} – u_{2})^{2} (1 – e^{2})
In Perfectly Inelastic One Dimensional Collision
Velocity of separation after collision = 0.
Loss of kinetic energy = m_{1}m_{2} (u_{1} – u_{2})^{2} / 2(m_{1} + m_{2})
If a body is dropped from a height h_{o} and it strikes the ground with velocity v_{o} and after inelastic collision, it rebounds with velocity v_{1} and rises to a height h_{1}, then
If after n collisions with the ground, the body rebounds with a velocity v_{n} and rises to a height h_{n} then
e^{n} = v_{n} / v_{o} = √h^{n} / h^{o}
TwoDimensional 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,
m_{1}u_{1} cos α_{1 }+ m_{2}u_{2} cos α_{2}= m_{1}v_{1} cos β_{1} + m_{2}v_{2} cos β_{2}
In vertical direction.
m_{1}u_{1} sin α_{1} – m_{2}u_{2} sin α_{2} = m_{1}u_{1} sin β_{1} – m_{2}u_{2} sin β_{2}
If m_{1} = m_{2} and α_{1} + α_{2} = 90°
then β_{1} + β_{2} = 90°
If a particle A of mass m_{1} moving along zaxis with a speed u makes an elastic collision with another stationary body B of mass m_{2}
From conservation law of momentum
m_{1}u = m_{1}v_{1} cos α + m_{2}v_{2} cos β
sin β_{2}v_{2} sin α – m_{1}v_{1}O = m
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