Collisions of Spheres

Introduction

            The following documents the calculation of final trajectories of colliding spheres.  For simplicity I will assume that the second sphere is not moving.  I will compute the trajectories of these spheres as a function of the fractional impact parameter which is defined as the offset between the line along which the center of the incoming sphere approaches and the center of the second sphere as shown in the diagram below.

Figure 1: Top shows the incoming sphere from the left.  Bottom shows the configuration at the moment of impact.  The radii of the spheres are a and b, respectively.

 

Calculations

There are some important parameters shown in Figure 1.  First, note that the momentum transfer, Dp, must be along the line of the radii a+b.  This line is at angle

                                                                                                               (1)

where o is the offset between centers shown in Figure 1. Also note that the momentum transferred to A is the negative of the momentum transferred to B:

                                                                                                                        (2)

The other requirements for this (assuming an elastic collision) are that the final total energies and momenta must be the same as the initial values (in this case, that of sphere A).

                                                                                                                     (3)

Equations 3 yield a useful relationship between the directions of pA and pB.  If we write out the expressions for energy and momenta in terms of the momenta, p, we have:

                                                                                        (4)

 We can take the dot product of both sides of the momentum vector equation and obtain:

                                                                   (5)

Combining the second line of equation 5 with the first line of equation 4 we must conclude that

                                                                               (6)

where q is the angle between pA and pB:

                                                                                           (7)

Also, because the only momentum that sphere B gets is along the line of centers (at angle a) we know that pA is at angle a-q with respect to the x axis.

Now let’s combine the geometry information we have just derived with the conservation equations to get the velocity vectors after the impact.  We can separate the momentum equation into x and y components as follows:

                                                                                   (8)

From the second of equations 8 we find the ratio pB/pA:                                                            

 

                                                                                                                 (9)

Then we use the ratio from equation 9 in equation 7 and obtain:

                                                                              (10)

which can be solved for q:

                                                                     (11)

Having found q, we can use equations 8 to find pA and pB in terms of pA0.

                                                                                                          (12)