Central Potential Repulsive Scattering

In summary, the conversation discusses the concept of total energy in a two-particle system and how it can be derived using the reduced mass. The speaker brings up a discrepancy in the equations when considering different frames of reference and concludes that energy conservation is not necessarily the same in every frame.
  • #1
ispivack
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0
Homework Statement
Consider two particles interacting via a repulsive central potential U(r) = k/r with k > 0. Find the minimal distance between particles, when one of them (with mass m1) is coming from infinity with initial velocity v0, and approaching an initially resting particle (with mass m2) with impact parameter ρ.
Relevant Equations
U(r) = k/r
I have one problem with this question that I've been struggling with. Initially, the total energy should be given by E =m1* v0^2/2 (as U goes to zero, and m2 is at rest). However, if we write r = r1 - r2, we get E = mu*rdot^2/2 + U_eff(r), U_eff(r) also goes to 0, where mu is the reduced mass. However, rdot^2 is exactly v0^2 when r is infinity, as rdot = r1dot - r2dot = r1dot, since m2 is at rest. This seems to imply that m1 = mu, which makes no sense. What am I missing? If I assume that the first energy I stated is incorrect, I have no trouble with the rest of the problem.
 
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  • #2
Simply because you have switched frames and illegally equated them. One frame is the 3rd observer,observing collision. In this frame,both particles are moving once they are sufficiently close to each other. Another frame is the frame of particle in which it is at rest and observing another particle coming towards him. These 2 frames coincide if the particle which was at rest, is too much heavy in which m=##\mu##.
 

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  • #3
When the particle m1 is infinitely far away, neither particle will have a force acting on it. So, both equations hold for the frame of the third observor, no?
 
  • #4
Why do you think this is true? If I only have a single particle is its KE the same for observers in various inertial frames? Manifestly not.
 
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  • #5
ispivack said:
When the particle m1 is infinitely far away, neither particle will have a force acting on it. So, both equations hold for the frame of the third observor, no?
Succintly,If you follow derivation of energy term using reduced mass(c.f. Goldstein), you see,you have to go into the frame of centre of mass. Now,conservation of energy is "Conservation" law not invariant law. It need not be same in every frame!
 
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Related to Central Potential Repulsive Scattering

1. What is central potential repulsive scattering?

Central potential repulsive scattering is a type of scattering phenomenon that occurs when a particle is deflected by a central potential force that repels it away from the center. This can happen in various systems, such as atoms, molecules, or nuclei, where the particles interact through a repulsive central potential.

2. How does central potential repulsive scattering differ from other types of scattering?

Central potential repulsive scattering differs from other types of scattering, such as elastic or inelastic scattering, in that it involves a repulsive central potential force that causes the particle to be deflected away from the center, rather than being scattered in different directions.

3. What factors affect the outcome of central potential repulsive scattering?

The outcome of central potential repulsive scattering is affected by several factors, including the strength of the repulsive force, the distance between the particles, and the initial energy and momentum of the particles.

4. What applications does central potential repulsive scattering have in science?

Central potential repulsive scattering has many applications in science, including in the study of nuclear reactions, the behavior of atoms and molecules, and the dynamics of gases. It is also used in various experimental techniques, such as scattering experiments and particle accelerators.

5. How is central potential repulsive scattering studied and analyzed?

Central potential repulsive scattering is studied and analyzed using various theoretical models and computational methods, such as the Born approximation and the scattering matrix method. Experimental techniques, such as particle detectors and spectroscopy, are also used to measure and analyze the outcomes of scattering events.

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