In a collision, involving two or more objects, when kinetic of the objects-system is not conserved but momentum is conserved, we term such a collision as Inelastic Collision.
In an inelastic collision the kinetic energy is lost as heat, sound, against friction etc of the objects involved.
In real world partial are the most common type of collisions. In this type of collision, the objects involved in the collisions do not stick together which happens in case of perfectly inelastic collision.
For a two body collision, let ##m_1and m_2## be masses of two bodies involved in the collision. Let ##v_(1i)and v_(2i)## be the initial velocities of the two bodies, and ##v_(1f)and v_(2f)## be final velocities of the colliding bodies after the collision.
General equations can be developed for the inelastic collision as below.
From the conservation of linear momentum we have
##m_1vecv_(1i)+m_2 vec v_(2i)=m_1vecv_(1f)+m_2 vec v_(2f)## ……(1)
This equation can be expressed as its corresponding three equations along the orthogonal Cartesian directions ##hatx, haty, hatz## and solved independently.
Let us introduce another equation which accounts for the inelastic nature of the collision. This is done using what is called the coefficient of restitution ##e##.
Line of impact ##L## passing through the centers of the colliding particles for inelastic collision is shown in the figure above.
The coefficient of restitution ##e## is given as
##e=(v_(L1f)-v_(L2f))/(v_(L1i)-v_(L2i))## …….(2)
Where ##v_(L1i)## is the component of the initial velocity of particle 1,
##v_(L2i)## is the component of the initial velocity of particle 2, ##v_(L1f)## is the component of the final velocity of particle 1, ##v_(L2f)## is the component of the final velocity of particle 2;
all above components resolved along the direction of ##L##.
Equations (1) and (2) can be solved simultaneously.
For the special case of a head on inelastic collision in one dimension, the coefficient of restitution reduces to
##e=(v_(1f)-v_(2f))/(v_(1i)-v_(2i))## ……….(3)
From equation (1) we have for one dimension,
##m_1v_(1i)+m_2 v_(2i)=m_1v_(1f)+m_2 v_(2f)## …..(4)
Equations (3) and (4) are then solved simultaneously.
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