**Subject :**Elements of Mechanical Engineering

## Direct Central Impact

**Direct Central Impact: **

Consider the collinear motion of two spheres of masses *m*_{1} and *m*_{2} travelling with velocities *V*_{1}and *V*_{2 }**.** If *V*_{1}** > ***V*_{2} **, **collision occurs with the contact forces directed along the line of centers. This condition is called direct central impact. Following initial contact, a short period of increasing deformation takes palce until the contact area between the spheres ceases to increase. At this instant, both the spheres are moving with the same velocity V_{0} **.** During the remainder of contact, a period of restoration occurs during which the contact area decreases to zero. In this period**,** mass *m _{2}* is applying a force on

*m*opposite to motion. Hence, the velocity of m

_{1}_{1}keeps on decreasing till the spheres separate out. Because of Newton's third law, the particle

*m*

_{1}aplies a force on

*m*

_{2}along the direction of motion. Hence the velocity of

*m*

_{2}keeps on increasing. Thus, after the impact velocity of

*m*

_{1}will always be lower than the velocit y of

*m*

_{2}.

In the elastic impact, the original shapes of the bodies are restored. In the inelastic impact, the shapes of bodies are not restored. In the perfectly plastic impact, bodies remain in contact and do not separate out.

Applying the law of conservative of linear momentum

The above equation contains two unknown to be determined and . Thus one more equation is needed to solve the impact problem. For this purpose **,** the coefficient of restitution is defined**.** The coefficient of restitution *e* is the ratio of the magnitude of the restitution impulses to the magnitude of the deformation impulses**.**

During deformation period

the force acting on each particle is F_{d }**.** During restitution period**,** the force acting on each particle is F_{r }**.**

For particle 1**,** the coefficient of restitution *e* is given as

Here from 0 to *t*_{0}, deformation takes place and from time *t*_{0} to *t* restoration takes place. *V*_{0 }is the velocity when both the particles are moving at the same velocity.