In mechanics, impact refers to when two bodies collide into each other, in a very short period of time they exert large impulsive force on to 
each other, therefore causing changes in their momenta, and hence their respective velocities.
How impact causes velocity changes is characterized by the coefficient of restitution between the two contacting materials.
An example of a field where impact is widely studied is sport. For example if you play tennis
you must take into consideration what court you're playing on: is it clay or grass or even carpet,
because you know that different tennis court will affect the bouncing of the tennis ball.
And this is because the coefficient of restitution between the two colliding materials, between the 
ball and the carpet, or between the ball and clay, or between the ball and grass, is all different.
Let's imagine when two particles are colliding into each other.
Their masses and initial velocities are known. During collision their contacting surface is known as the plane of contact.
And the line that is perpendicular to the plane of contact
is the line of impact.
In this case since both velocities of the particles are along the line of impact, there is no velocity
component that is at an angle to the line of impact. This case is known as central impact.
Now the question is what are the velocities of the two particles after the collision.
To solve this problem we will treat the two particles as one system.
The impulsive forces they exert on each other are internal. There is no external impulsive force 
acting on the system. Therefore we can apply the conservation of linear momentum on the system.
As you can see we have one equation
but two unknowns: the final velocities of the two particles.
Therefore we cannot solve for both unknowns unless we have another equation.
Where should we find the other equation?
Let me remind you though: you shouldn't readily assume that the energy of the system is also conserved.
If we do know the coefficient of restitution, e, between the two colliding materials, then we will have
one more equation, and we will have two equations and we can solve for the
two unknowns.
The coefficient of restitution equals to the difference in the final velocites v_B_2 minus v_A_2, over the difference in the initial velocities
v_A_1 minus v_B_1. Notice here you must switch the order of the two particles on the numerator and the denominator.
Just like the coefficient for friction, coefficient of restitution e is generally determined experimentally.
The value is normally between zero and one.
A value of one refers to a so-called perfect elastic impact which means that the two 
colliding particles would quickly restore to their original shape and size after the initial
deformation caused by collision.
There is no energy lost to deformation or sound or other type, and the total energy is conserved for the system, 
and the two particles will separate at the same relative speed as they were approaching.
And this is only in theory.
A coefficient of restitution
that equals to zero indicates plastic impact: this is when the two colliding particles experience the largest deformation and they would
stick togather and move at the same speed. Their relative separation velocity is zero.
In this case the energy loss is maximum.
Let's look at this example. Although the two particles are moving both horizontally, when they collide into each other their plane of contact is 
at a 45° angle with the horizon, so in this case there exists velocity component that is at an angle to the line of impact.
And unlike the central impact that we introduced earlier, this is an example of oblique impact
And we need to determine the respective speeds after the collision of these two particles
and we know that the coefficient of restitution between them is 0.5.
So first we draw the line of impact that is perpendicular to the plane of contact.
And now it is more convenient for us to set up our x-y coordinate system this way with x axis along the line of impact and
y axis perpendicular to the line of impact. And now we can
resolve the two velocities into their respective x and y components.
And by applying trigonometry we can quickly determine the values of these velocity components.
Since the x axis is set up to be along the line of impact, x direction is where impact happens, so along the x direction we can apply 
the conservation of linear momentum by treating the two particles as one system.
And their total linear momentum before the collision
is the same as their final
total linear momentum after the collision
along the x axis. 
And since we know that the coefficient of restitution is 0.5,
we have two equations, two unknowns,
therefore we can solve for both unknowns, which are the final velocities of the two particles, but only along the x direction.
We still need to know the final velocities of the two particles along the y direction.
Don't forget the y direction is perpendicular to the line of the impact.
For the y direction we can apply the principle of linear impulse and momentum to each individual particle,
for particle A and for particle B. However along the y direction there is no force,
therefore there is no linear impulse for either particle.
Therefore,
for each particle the final velocity along the y direction is simply the same as the initial velocity along the y direction.
And now since we know the x and y components of the final velocities of these two particles after the collision, we can easily 
determine the magnitude of their respective finial velocity.
And that completes this problem,
but if you'd like to you can also calculate the direction of the two velocities.
