Consider first the case of a particle of mass m, moving on a smooth horizontal surface with speed u, towards a fixed block whose face is perpendicular to the direction of motion of the particle. When the particle hits the block an impulse J is exerted on the particle by the block and, if the impact is elastic, the particle bounces off the block in the opposite direction with speed v, say.
The approach speed is u, and the separation speed is v.
Therefore using the law of restitution gives v = eu
Now, taking the direction of J as positive,
Gives J = mv – (-mu)
These are the two principles that can be applied to situations of this type in which one of the colliding objects is fixed. The conservation of linear momentum is not valid in such cases for the impulse applied to the particle by the fixed surface is an external impulse; hence the momentum of the particle is changed but the momentum of the fixed object is not changed by an equal and opposite amount.
The principle of conservation of linear momentum can be applied to the direct impact of particles both of which are free to move, as this is a case where the pair of impulses at impact cause equal and opposite changes in momentum and so have no overall effect on the total momentum of the system.
For this situation therefore, in addition to the law of restitution, we can also use conservation of momentum.
Because we will now be dealing with more than one moving particle, it is particularly important to choose a positive direction when dealing with the momentum of the system.
MULTIPLE IMPACTS
Sometimes a collision between two objects leads to further collisions either wit another moveable object or with a fixed surface. In such cases impact can be dealt with by the methods already described. It one collision completely before starting on the next one, and with a new set of diagrams. The positive direction can be chosen afresh for each collision – it need not be the same throughout.
All the collision so far considered have been direct, i.e. the impulse that acts on a sphere at impact is in line with the direction of motion of that sphere both before and after the impact.
We now consider a situation where the impact is not direct but oblique.
Suppose that when a sphere traveling on a horizontal surface vertical wall, the direction of its velocity makes an angle A with the wall.
The impulse exerted on the sphere is perpendicular to the wall and change in the momentum of the sphere in that direction; it does not affect the momentum parallel to the wall.
So if the approach velocity of the sphere is resolved into components parallel and perpendicular to the wall, one of these components is changed by the impact and the other is unchanged.
It follows that the direction, as well as the magnitude, of the velocity of the can be changed by the impact.
Oblique Collision Between Two Moving Objects
if two spheres, of equal radii, are free to move on a horizontal collide when their velocities are not in the same straight impulses that act on impact are perpendicular to the common tangent spheres and so lie on the line joining the centres of the spheres.
Therefore, for each sphere there is a change in momentum (and hence speed) along this line of centres but not perpendicular to it.
The unchanged components can be incorporated in a working diagram; then along the line of centres the calculation is exactly the same as for direct impact, i.e. conservation of momentum and the law of restitution can be applied in this direction.
Note that the line of centers is horizontal because the spheres have equal radii.
