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

[lriuui0x0]

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.

 

[bob012345]

You have used energy conservation. What else can you use?

 

[lriuui0x0]

Thanks. I solved the problem by noticing another relationship $l = pw$.

 

[bob012345]

Thanks. I solved the problem by noticing another relationship $l = pw$.

Exactly, conservation of angular momentum $l = bu = pw$.