Central force field derivation

In summary, the total energy of a particle can be calculated using the formula ##u^2 / 2 - k/R##. If ##u^2 \gg 2k/R##, we only use the first part of the formula, ##u^2/2##. By applying the conservation of energy law, we can use the angular momentum expression ##l = bu## to replace ##u## in the formula and obtain an expression for ##w^2 / k## using only the variables ##b, p, l##. Another helpful relationship is ##l = pw##, which can be used to solve for the problem.
  • #1
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Homework Statement
A particle with unit mass has distance ##R## from the origin and initial speed ##u##. It moves in the central potential ##\Phi(r) = -k/r##. If it doesn't move in the central field, it would move in a straight line whose shortest distance from the origin is ##b## (impact parameter). When it moves in the central field, it's closest distance from origin is ##p < b## with speed ##w##. Assume ##u^2 \gg 2k/R##, find ##w^2/k## in terms of ##b, p## only.
Relevant Equations
##l = bu##
The total energy of the particle is ##u^2 / 2 - k/R##. When ##u^2 \gg 2k/R##, we take the total energy to be ##u^2/2## only. By the conservation of energy, we have:

$$
\frac{u^2}{2} = \frac{w^2}{2} - \frac{k}{p}
$$

Take the angular momentum expression ##l = bu##, we can replace ##u## with ##b,l## thus getting an expression for ##w^2 / k## with ##b,p,l## only. But I don't know how to get an expression with ##b, p## only.
 
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  • #2
You have used energy conservation. What else can you use?
 
  • #3
Thanks. I solved the problem by noticing another relationship ##l = pw##.
 
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  • #4
lriuui0x0 said:
Thanks. I solved the problem by noticing another relationship ##l = pw##.
Exactly, conservation of angular momentum ##l = bu = pw##.
 
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