mv − ∫ Fdt = S (Eq. 1)

^t0

Where v and v₀ is the speed after and before the collision, F is the impact force; S is the impact impulse and τ is the duration of the collision. The left-hand side of this equation is a finite value and therefore, the impulse S is a finite value, too. The integral in the above equation could be replaced by the product of F^*.τ, where the force F^* should obviously be very high [19]. This statement could also be confirmed through the following arguments:

In order for the amount of movement of a certain material particle to be varied by a finite value within a very short time interval it should be subjected to the action of a very strong force. The force F^* which is acting during the collision and reaches very high values is called impact force and its impulse is called impact impulse.

7.1.2 Displacement of particles during the collision

Practice has proven that collision duration is within 10^-3 and 10^-6 s and as the speeds before and after the collision are finite values, the displacement during the collision will therefore be negligible.

7.1.3 Physical grounds of collision

From all described above we can confirm that the collision of two bodies causes strong impact forces to occur, the value of which exceeds regular forces many times. In order to explain the physics of these forces it will be necessary to assume that the colliding bodies are deformable. Experience shows that whenever there is collision between two bodies they undergo deformation, which occurs for a very short time interval and completely or partially disappears just as quickly [5].

If we followed the collision of two spheres whose speeds lie on the straight line running between their centres, we would find that the two bodies start to deform from the moment of the collision. Their speeds become very close in value and the distance between their centres tends to become smaller than the sum of their radii. It is assumed that the end of this stage, which is also referred to as the deformation stage, is the moment when their speeds become equal and the distance between their centres becomes the minimum possible [5].

Following immediately after that is the restitution stage when an irreversible process starts. The two bodies start to restore their shapes to one or another extent depending on their elastic characteristics. The restitution stage is complete when the two bodies separate.

If deformation disappears completely after the collision this is referred to as a perfectly elastic collision. If deformation is completely retained after the collision this is referred to as a perfectly non-elastic (non-plastic) collision [20]. In the case of the latter the two bodies do not separate after the collision but start moving as one body. For actual real bodies deformation usually partially disappears and is partially retained. In this case the collision is referred as imperfectly elastic.

7.1.4 Basic assumptions

a) When a point or a mechanical system is subjected to the action of regular forces their value during the collision is disregarded. The ground for this lies in the fact that their impact impulses are negligible.

b) The displacement of material objects during the collision is disregarded; i.e. bodies are assumed to be stationary.

 

7.2 Collision of mechanical objects
7.2.1 Material particle collision

 

a) Basic equation for the dynamics of a particle during a direct collision.

If a particle having a mass m is subjected to an impact force at a certain moment of time, then based on the theorem for the variation of the amount of movement we can write down the following equation:

m .u − m.v = S (Eq. 2) where v is the speed of this particle before the collision and u is the speed of the same particle after the collision. This equation is called the basic equation for the dynamics of a particle during collision. Considering the assumptions we made in the previous paragraph, the action of the impact force can be evaluated by its impact impulse, which in the impact theory plays a role that is similar to regular forces acting during the movement of material objects [20].

b) A particle colliding in a stationary surface

Let us consider a falling material particle colliding in a stationary surface. The task is to determine the speed of this particle u after the collision and its direction β knowing the mass m of the particle, the speed v before the collision and the direction of the impact relative to the normal drawn to the colliding surface α.

The following two Newton’s laws could be used to give a satisfactory answer to these questions:

− The ratio of the absolute values of the normal projections of the relative speeds of the particle after and before the collision is a constant value, which is independent of the relative speeds nor of the size of the colliding bodies and is dependent just on their physical properties [20].

u
u
n

 

−
k

 

, (Eq. 3) v_n
ⁿ= k_{or vn}

 

Where u_n and v_n are the projections of the speed after and before the collision on the normal drawn to the collision surface and k is the restoration factor.

− The ratio of the tangential projections of the particle relative speed after and before the collision is a constant value, which is independent on the relative speeds and size of the colliding bodies and is only dependent on the nature of the material they are made of:

^uτ − λ, (Eq. 4) _vτ

 

Where λ is a very negligible value and is called coefficient of momentary friction. These two laws help me derive the relation for the value of the speed after the collision: