I Maximum impact parameter given effective potential

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The discussion revolves around understanding why higher particle energies lead to capture in gravitational interactions, contrary to the intuition that lower energies would result in capture due to insufficient energy to escape. The key point is that the condition for capture is defined by the effective potential, where the maximum effective potential must be less than or equal to the particle's energy (V_eff^max ≤ E). This relationship indicates that as energy increases, the impact parameter decreases, resulting in a smaller cross-section for capture. Additionally, higher energy correlates with increased angular momentum, which further influences the dynamics of capture. The explanation clarifies the misconception about energy and gravitational capture in classical mechanics.
stephen8686
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This problem is from David Morin's classical mechanics textbook:
problem.PNG

I am having trouble with Part b. Here is the textbook's answer:
andswe.PNG


I do not understand why large particle energies lead to capture. I would think that smaller energies would lead to capture because the particle wouldn't have enough energy to escape the gravitational potential, whereas large energy particles could woosh past. If someone could explain why my intuition is wrong, that would be very helpful.
 
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How do you make out that larger energies lead to capture?
 
PeroK said:
How do you make out that larger energies lead to capture?
That's what the answer says, "The condition for capture is therefore ##V_{eff}^{max}\leq E## " That is the part of the answer that I don't understand
 
stephen8686 said:
That's what the answer says, "The condition for capture is therefore ##V_{eff}^{max}\leq E## " That is the part of the answer that I don't understand
That condition resolves into a smaller impact parameter and smaller cross section for capture for greater energy.
 
stephen8686 said:
That's what the answer says, "The condition for capture is therefore ##V_{eff}^{max}\leq E## " That is the part of the answer that I don't understand
That equation in itself is about the relationship between angular momentum and energy. But, angular momentum increases with energy if other factors are held constant, so it doesn't say what you are thinking it says.
 
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