# Does inelastic collision mean the colliding particles have to necessarily stick?

In a case when particle 1 is moving vertically upwards and particle 2 if moving horizontally and they collide perfectly inelastically, why would the particles stick ?

Since inelastic collision means coefficient of restitution = 0, only their horizontal velocities should become equal, since there is no impulse in the vertical direction to change the individual particles momentum. But in a question in my book, they have assumed the two particles stick together and move together. I don't understand why would that happen.

• Kinetic energy is not about direction of velocity but only about the speed. For a perfectly inelastic collision all this initial kinetic energy is "spent" and transformed in the collision. – Steeven Jan 30 '16 at 18:48
• See Wikipedia. – Qmechanic Jan 30 '16 at 19:01
• @Steeven Minor nitpick: your statement is true in the center of momentum frame, but not in others where it is all the kinetic energy associated with motion relative the CoM that is converted to other channels. – dmckee --- ex-moderator kitten Jan 30 '16 at 19:13
• Elastic and completely inelastic are two extreme ends of a spectrum. They are idealizations, and neither can be obtained in real life. Inelastic may refer to completely inelastic or incompletely inelastic. Which it is should be stated, or be clear from context. When two particles stick the collision is nearly completely inelastic. Why do two objects stick? That's a function of the material and chemistry. Some do, some don't. – garyp Jan 30 '16 at 20:05
• Since particle 1 is moving vertically, it provides a vertical impulse to particle 2. Symmetrically, particle 2 provides a horizontal impulse to particle 1. Combined with their initial velocities, both particles now move diagonally in the same direction. – user854 Jan 31 '16 at 3:02

## 1 Answer

No it is not necessary that the colliding particles 'stick' together after a totally inelastic collision, in the sense that a force must be applied to separate them again. Some kinetic energy is lost, which could be used to 'fuse' the two particles or form a bond between them. But this energy could otherwise be transformed into heat or permanently deform the particles without making them 'stick' together.

The essential requirement is that the relative velocity of separation becomes zero - ie the particles move together with the same velocity after the collision. Alternatively, "totally inelastic" means that in the centre of momentum frame of reference (in which the centre of mass of the particles is at rest) all of the kinetic energy is destroyed by the collision, transformed to internal energy.

This applies in all 3 dimensions. You cannot have a collision which is totally inelastic in one direction and only partially inelastic in another direction, because then the relative velocity after the collision would not be zero, which contradicts the definition of a "totally inelastic" collision.

In your problem, if after colliding the particles have the same velocity in the x-direction but different velocities in the y-direction then the relative velocity after the collision is not zero, so by definition the collision was not totally inelastic.