Centre of Mass frame of colliding molecules

In summary, Feynman means that the relative velocity of the particles before the collision is not associated with the relative velocity of the particles after the collision.
  • #1
jbunten
87
0
I'm not sure this is the right place to post this question but here goes:

In feynman vol.1 39-9 there is a situation where two molecules are about to collide in a CM frame, the frame has velocity Vcm and the two molecules have respective velocities v1 and v2, then there is no correlation between Vcm and the relative velocity, w (w=v1-v2) "coming in". what does Feynman mean by "coming in"?

I find this a very difficult argument to understand, specifically, is w really relative velocity *before* the collision? in that case how can it be independent of Vcm, surely if one of the molecules is very heavy and very fast and the other slow and light, then Vcm is moving very fast in roughly the same direction of the heavy molecule and a correlation exists.

If someone could look this up I'd appreciate it as I've been trying to get my head around it for ages, many thanks.
 
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  • #2
jbunten said:
I'm not sure this is the right place to post this question but here goes:

In feynman vol.1 39-9 there is a situation where two molecules are about to collide in a CM frame, the frame has velocity Vcm and the two molecules have respective velocities v1 and v2, then there is no correlation between Vcm and the relative velocity, w (w=v1-v2) "coming in". what does Feynman mean by "coming in"?
.

My copy of Feynman, Leighton, and Sands (1963) has the
discussion on pages 39-8 and 39-9. Maybe you have a more
recent version, in which case some of what I say may not
be appropriate.

The presentation in my copy is rather confusing and the
diagram (Fig. 39-3) is incorrect: there are two different
vectors with the same label v1 and two
different vectors with label v2. In
addition, the caption says "...viewed in the CM system",
whereas the text defines the vectors v to be in the
laboratory system.

In what follows, capital letters refer to vectors in
the laboratory system and lower case letters refer
to the centre-of-mass (CM) coordinate system or to the
coordinates of one particle relative to the other. I am
going to consider the two-body problem where there are
no external forces and where the internal forces are
central forces obeying Newton's third law.

Let Ri and Vi be the position and
velocity vectors respectively of particle i in the
laboratory system and let mi denote the mass of particle i.
Introduce two new coordinates:

R = (m1R1+m2R2)/M
r = R1 - R2
M = m1 + m2

The inverse transformation is

R1 = r + m2r/M
R2 = r - m1r/M

There are similar equations for the velocities, obtainable
from the above by differentiating wrt time. In the absence
of external forces, the transformation achieves a separation
of the two-body problem into two single-particle problems:

M dV/dt = 0
m dr/dt = F1i
m = m1m2/M

where F1i is the force exerted by
particle 2 on particle 1. and m is the reduced mass.
You can see from this that the
centre-of-mass travels in a straight line with constant
velocity, regardless of the force acting between the
particles. That is what is meant by "uncorrelated", nothing
more. When Feynman says "coming in" he means the relative
velocity before the collision.

To see what happens after the collision, you must see two
things: the linear momentum is MV both before and
after the collision, and the kinetic energy is

T = 1/2(MV2 + mv2)

For an elastic collision, the first condition will
be satisfied if the velocities of
particles 1 and 2 after the collision as seen from the
centre-of-mass are in opposite
directions and the second condition will be satisfied if
the magnitude of v is the same before and after
the collision. This means that the CM velocity vector
for particle 1 must lie on a sphere centred at the end
of the vector V and the CM velocity vector for
particle 2 must be in the opposite direction and
ending on a second sphere centred at the end of V.

To "get your head around this" draw a Newton diagram:
Draw vectors V1 and V2
with their tails placed in the origin of the laboratory
system. Draw vector v. The vector V has its
tail in the laboratory origin and its head somewhere on the
vector v---the exact position depends upon the masses.
Using the end of this vector as a new origin, draw a circle
of radius m2v/M showing all possible
velocities of particle 1 after the collision. Draw another
circle of radius m1v/M showing the
possible velocities of particle 2 after the collision.

Hope this is of some help. You have to stare at the diagram
a while.
 
  • #3
Thanks for your reply and sorry for taking a while to answer, that was a really good and detailed explanation. I think my confusion arose from the poor definition of "uncorrelated" and the confusing diagram. Once I drew the diagram as you said it made perfect sense.

Many thanks.
 

1. What is the centre of mass frame of colliding molecules?

The centre of mass frame of colliding molecules is a reference frame in which the total momentum of the colliding molecules is zero. This reference frame is useful for studying the dynamics of molecular collisions and can provide insights into the energy and momentum transfer between the molecules.

2. How is the centre of mass frame determined for colliding molecules?

The centre of mass frame is determined by finding the point where the total momentum of the colliding molecules is zero. This can be calculated by taking the weighted average of the positions and velocities of the molecules, where the weights are determined by the masses of the molecules.

3. Why is the centre of mass frame important in studying molecular collisions?

The centre of mass frame allows us to simplify the analysis of molecular collisions by removing the effects of overall translation and rotation of the molecules. This allows us to focus on the internal dynamics of the collision, such as energy and momentum transfer between the molecules.

4. How can the centre of mass frame be used to study the properties of colliding molecules?

By studying the energy and momentum transfer between the molecules in the centre of mass frame, we can gain insights into the properties of the molecules, such as their masses and collision cross-sections. This information can be used to better understand the behavior of molecules in various environments.

5. Can the centre of mass frame be applied to all types of molecular collisions?

Yes, the concept of the centre of mass frame can be applied to all types of molecular collisions, as long as the total momentum of the colliding molecules is conserved. This includes collisions between gas molecules, liquid molecules, and even molecules in a solid state.

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