Relative velocity vector

Fig. 24. YC3H4 and YCH2 product flux distributions from collisions of Y + propene at Boon = 12.3kcal/mol and 25.2 kcal/mol, respectively. Arrows represent the CM relative velocity vectors for the colliding reactants.

The maj or limitation of the TAB model i s that it can only keep track of one oscillation mode, while in reality there are many oscillation modes. Thus, more accurately, the Taylor analogy should be between an oscillating droplet and a sequence of spring-mass systems, one for each mode of oscillations. The TAB model keeps track only of the fundamental mode corresponding to the lowest order spherical zonal harmonic 5541 whose axi s i s aligned with the relative velocity vector between the droplet and gas. Thi s is the longest-lived and therefore the most important mode of oscillations. Nevertheless, for large Weber numbers, other modes are certainly excited and contribute to droplet breakup. Despite this... 

Let us consider b as the impact parameter, i.e. distance between the centre mass of B and relative velocity vector as shown in Fig. 9.2. The contribution of impact parameter from b to b + db to the collision cross section is... 

To specify and define the initial state of collision, z-axis may be selected as the direction of the initial asymptotic relative velocity vector uR as shown in Fig. 9.22. 

Consider the three collisions depicted in Fig. 10.3. In the collision at the top of the figure, the relative velocity vector is aligned directly between the centers of the two molecules. As such, they undergo a head-on collision. In this model all of the translation energy would be available, if needed, for passing over the reaction barrier. 

Figure 4.59. Distribution of the relative velocity vector in the front region during mold filling.

The dynamics of the two-particle problem can be separated into center-of-mass motion and relative motion with the reduced mass /i = morn s/(rnp + me), of the two particles. The kinetic energy of the relative motion is a conserved quantity. The outcome of the elastic collision is described by the deflection angle of the trajectory, and this is the main quantity to be determined in the following. The deflection angle, X, gives the deviation from the incident straight line trajectory due to attractive and repulsive forces. Thus, x is the angle between the final and initial directions of the relative velocity vector for the two particles. 

Fig. 4.1.7 Two-body classical scattering in a spherically symmetric potential. The relative motion of two atoms may be described as the motion of one pseudo-atom , with the reduced mass p = m-Ams/(rnA+rn-s), relative to a fixed center of force. Two trajectories are shown for the first trajectory, the final and initial relative velocity vector and the associated deflection angle x are shown. This trajectory corresponds to the impact point (b, = 0), whereas the second trajectory corresponds to the impact point (b, 4> = ir).

We have considered above the deflection of the trajectory in the center-of-mass coordinate system. The deflection angle x is the angle between the final and initial directions of the relative (velocity) vector between the two particles. Experimental observations normally take place in a coordinate system that is fixed in the laboratory, and the scattering angle 0 measured here is the angle between the final and the incident directions of the scattered particle. These two angles would be the same only if the second particle had an infinite mass. Thus, we need a relation between the angles in the two coordinate systems in order to be able to compare calculations with experiments. 

Vj Relative velocity vector between barrel surface and solid bed (9.3-1)... 

All the experiments described in this chapter were performed on molecules in gas cells, and, unlike in beam experiments, the results obtained are averaged over all possible orientations of the molecules with respect to the relative velocity vector, and over the Maxwellian distribution of the relative velocities of the colliding particles. Nevertheless, the experimental data stimulated interest in the development of theoretical models of elastic... 

Two such frames that will be used are those with the z axis directed perpendicular to the collision plane for the standard collision and with the z axis along the relative velocity vector before the collision. The z projections of j in these two frames are denoted by m and n ft stands for the projection of j on the rotating molecular axis. 

Note that for two spherical molecules with central force fields, the relative motion will take place in a plane. This plane will be defined by the centers of the two molecules and the direction of their relative velocity vector, which is parallel to the plane. No forces act perpendicularly to the plane. 

The magnitude of the final relative velocity vector p is determined by the energy conservation restriction... 

Moreover, the average relative velocity vector (10.105) is approximated by (10.113), as for the CPV model, because the drift velocity is neglected. 

The transformation matrix from the laboratory frame (x) into the collision frame of reference (x ) depends only on the relative velocity vector V12 = vi - V2. By convention, V12 is aligned with the x axis. Thus, the orthonormal transformation matrix defined such that x = Lx can be written as... 

Thus the rate of change of the distance between two neighboring material points depends on only the rate-of-strain tensor E, i.e., on the symmetric part of Vu. It can be shown in a similar manner that the contribution to the relative velocity vector Su that is due to the... 

The dependence of chemical reaction on the orientation of the reactant molecule with respect to the initial relative velocity vector may be studied by... 

In our experiments we also measured the velocity component along the initial relative velocity vector of the product H+ ion. The results showed that all H+ ions... 

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