Collisions and Scattering

In our analysis of the swinging ball machine we discussed the elastic collision of two spherical objects when the collision was head on, that is the velocities of the two objects were along the line connecting the objects' centers. In order to understand the scattering of a group of light particles by a group of heavy ones, we must extend our analysis into more than one dimension.

Let us consider the case when the ball 1 collides the ball 2 without the friction. We shall suggest that the ball 2 of mass m2 is in the rest before collision and the ball 1 of mass m1 is moving with velocity v. The velocity v1 of ball 1 and velocity v2 of ball 2 after collision will depend upon the "aiming" distance d , which is equal to the distance between the center of the ball 2 and the line of the motion of the ball 1 before collision. The collision will happen if d < r1 + r2, where r1 and r2 are the radiuses of the ball 1 and ball 2 consequently. The force applied to ball 2 during collision from the side of the ball 1 is directed along the line joining the centers of the balls. So, after collision the ball 2 will move at angle q as shown in the figure.

(r1 + r2) sin q = d

During the collision the energy and momentum of the motion is constant:

From these equations we can find

If m1<<m2 the picture of the momentum of balls is shown in the figure (p=mv, p1=mv1, p2=mv2). We can see from this figure that in the case of frontal collision (q = 0) p2> 2p, p1> -p. In any case the velocity of the ball 1 is changed by the direction, not by the value. The velocity of the heavy ball 2 after collision will not exceed the value v2=2vm1/m2.

Now we can move on to the situation illustrated. Here we have a number of light particles striking a number of heavier ones. There are differences between our analysis so far and this case. One is that multiple light particles may hit the same heavy one and the heavy particle is not so heavy that its reaction to collision may be neglected. This means that the heavy particle may not be at rest when the second and subsequent light particles hit it. Another difference is that a single light particle may rebound from one heavy particle and strike a second heavy particle. These differences enormously complicate the analysis. About all we can say at this point is that the scattering angles of the heavy particles will be in the range -p /2 to from +p /2 from the initial velocity vector of the light particles and this will bias the scattering of the group, introducing a preferential forward scattering. Some of the light particles may indeed be scattered into the back hemisphere, but more of them will be scattered into the forward hemisphere.