Kinematics of the elastic collision

Let us consider two point-like particles with masses $m_1$ and $m_2$, coordinates $\mathbf{r}_1$ and $\mathbf{r}_2$ and velocities $\mathbf{v}_1$ and $\mathbf{v}_2$. One can introduce the center of mass vector $\mathbf{r}_c$:

$$\mathbf{r}_c=\frac{m_1\mathbf{r}_1+m_2\mathbf{r}_2}{m_1+m_2}$$ (2.1)

and the relative position vector:

$$\mathbf{r}=\mathbf{r}_1-\mathbf{r}_2.$$ (2.2)

Their time derivatives are: the velocity of the center of mass:

$$\mathbf{v}_c=\frac{m_1\mathbf{v}_1+m_2\mathbf{v}_2}{m_1+m_2}$$ (2.3)

and the relative velocity:

$$\mathbf{V}_{12}=\mathbf{v}_1-\mathbf{v}_2.$$ (2.4)

The forces between these two particles depends only on their relative position and are of equal magnitude and pointing in opposite directions:

$$\mathbf{F}_{12}(\mathbf{r})=-\mathbf{F}_{21}(\mathbf{r}).$$ (2.5)

This is equivalent to say that the center of mass does not accelerate, i.e.:

$$\frac{d^2 \mathbf{r}_c}{dt^2}=0$$ (2.6)

while the relative position obeys to the following equation of motion:

$$m^*\frac{d^2\mathbf{r}}{dt^2}=\mathbf{F}_{12}(\mathbf{r})$$ (2.7)

where

$$m^*=\left(\frac{1}{m_1}+\frac{1}{m_2} \right)^{-1}$$ (2.8)

is the reduced mass of the system of two particles. If the collision is elastic an interaction potential can be introduced so that:

$$\mathbf{F}_{12}=-\frac{dU(r)}{dr}\hat{\mathbf{r}}$$ (2.9)

where $\hat{\mathbf{r}}$ is the unit vector along the direction of the relative position of the two particles. The force vector lies in the same plane where the relative position vector and relative velocity vector lie. The evolution of the relative position $r$ is the evolution of the position of a particle of mass $m^*$ in a central potential $U(r)$. The angular momentum of the relative motion $\mathbf{L}=\mathbf{r} \times m^* \mathbf{V}_{12}$ is conserved. This means that the particle trajectory, during the collision, will be confined to this plane. In Fig. 2.1 is sketched the typical binary scattering event when the interacting force is repulsive (monotonically decreasing potential), in the center of mass frame.

Figure 2.1: The binary elastic scattering event in the center of mass frame, with a repulsive potential of interaction

In the center of mass frame the elastic scattering has a very simple picture: the velocities of the particles are $\mathbf{v}_{1c}=\mathbf{V}_{12}m^*/m_1$ and $\mathbf{v}_{2c}=-\mathbf{V}_{12}m^*/m_2$. The elastic collision conserves the modulus of the relative velocity $V_{12}$ and therefore also the moduli of the velocities of the particles in the center of mass frame. If one consider the collision event as a black box and observes the velocities of the particles ``before'' and ``after'' the interaction (i.e. asymptotically, when the interaction is negligible), then the velocity vectors are simply rotated of an angle $\chi$ called angle of deflection, which also represents the angle between asymptotic initial and final directions of the relative velocity. During the collision the total momentum is conserved (this happens also for inelastic collisions) but is redistributed between the two particles, i.e. the variation of the momentum of the particle $1$ is $\delta(m_1 \mathbf{v}_1)=m^*(\mathbf{V}_{12}'-\mathbf{V}_{12})$ where the prime indicates the post-collisional relative velocity. Obviously $\delta(m_1\mathbf{v}_1)=-\delta(m_2\mathbf{v}_2)$. Finally, one can calculate the components of the momentum transfer parallel and perpendicular to the relative velocity:

To calculate the angle of deflection $\chi$ one needs the exact form of the interaction potential, the asymptotic initial relative velocity $V_{12}^0$ (i.e. at a distance such that the interaction is negligible) and the impact parameter $b$ that is the minimal distance between the trajectories of the particles if there were no interaction between them:

$$\chi=\pi-2\int_{r_m}^\infty dr\frac{b}{r}\left[ r^2-b^2-\frac{2r^2U(r)}{m^*(V_{12}^0)^2} \right]^{-1/2}$$ (2.11)

where $r_m$ is the closest distance effectively reached by the two particles. From Eq. (2.11) it is evident that the angle of deflection decreases as the initial relative velocity increases.

The case of an inverse power potential is of physical interest:

with $a>1$ and $K_a$ positive or negative in order to have respectively repulsive or attractive forces. In this case Eq. (2.11) can be explicitly calculated for some specific values of the power $a$. For example the gravitational or electrostatic interactions (in 3D) correspond to the case $a=2$ for which

$$\chi=2 \arcsin \left[ \frac{1}{\sqrt{1+x_0^2}} \right]$$ (2.13)

where $x_0=b(m^*(V_{12}^0)^2/K_2)$. The case $a=3$ is even simpler leading to the formula

$$\chi=\pi \frac{\sqrt{b_0^2+b^2}-b}{\sqrt{b_0^2+b^2}}$$ (2.14)

with $b_0=(K_3/m^*(V_{12}^0)^2)^{1/2}$. The case $a=5$ (in 3D) is of particular interest because for this special interaction the kinetic equation (Boltzmann equation, see section 2.2.3) is very simple. Particles interacting with this potential are called Maxwell molecules, discussed in more detail in paragraph 2.2.5. The deflection angle for this case is quite complicated:

$$\chi=\pi-\int_0^1du\frac{2^{3/2}x_0\sqrt{\sqrt{x_0^4+2}-x_0^2}}{\sqrt{(x_0^4+2)-\left\{x_0^2+u^2\left[\sqrt{x_0^4+2}-x_0^2 \right] \right\}^2}}$$ (2.15)

with $x_0=b(m^*(V_{12}^0)^2/K_5)^{1/4}$. For small values of $x_0$ (i.e. for small impact parameters or small initial relative velocities) the deflection angle approaches $\pi$ linearly, i.e.:

$$\underset{x_0 \to 0}{lim} \chi=\pi-3.118x_0.$$ (2.16)