How can one derive an expression for \( w^2 / k \) using \( b, p, l \) only?

AI Thread Summary
The discussion focuses on deriving an expression for \( w^2 / k \) using the variables \( b, p, l \). It begins with the total energy of a particle, expressed as \( u^2 / 2 - k/R \), and simplifies to \( u^2 / 2 \) under certain conditions. By applying conservation of energy, the relationship \( \frac{u^2}{2} = \frac{w^2}{2} - \frac{k}{p} \) is established. The angular momentum equation \( l = bu \) allows for substitution of \( u \) in terms of \( b \) and \( l \). The problem is ultimately solved by recognizing the relationship \( l = pw \), linking all variables effectively.
lriuui0x0
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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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You have used energy conservation. What else can you use?
 
Thanks. I solved the problem by noticing another relationship ##l = pw##.
 
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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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