Maximum impact parameter given effective potential

In summary, the conversation discusses the concept of capture in classical mechanics and the confusion regarding the relationship between particle energy and capture. The textbook's answer explains that the condition for capture is determined by the maximum effective potential energy being less than or equal to the particle's energy. This results in a smaller impact parameter and cross section for capture as energy increases. However, the equation mentioned only describes the relationship between angular momentum and energy, and does not support the idea that larger energies lead to capture.
  • #1
stephen8686
42
5
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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  • #2
How do you make out that larger energies lead to capture?
 
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  • #3
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
 
  • #4
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.
 
  • #5
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.
 

Related to Maximum impact parameter given effective potential

1. What is the maximum impact parameter?

The maximum impact parameter is the largest distance at which a particle can approach a central force without being captured or deflected.

2. How is the maximum impact parameter calculated?

The maximum impact parameter is calculated using the effective potential, which takes into account the attractive and repulsive forces acting on the particle.

3. What does the maximum impact parameter tell us about the particle's trajectory?

The maximum impact parameter determines the closest distance that a particle can come to the central force before being deflected or captured. It can also indicate the shape of the particle's trajectory, such as a circular or elliptical orbit.

4. How does the maximum impact parameter change with different central forces?

The maximum impact parameter is dependent on the strength and direction of the central force. As the central force increases, the maximum impact parameter decreases, meaning the particle must approach closer to the center before being deflected or captured.

5. What are some real-world applications of the maximum impact parameter?

The maximum impact parameter is used in various fields such as astrophysics, celestial mechanics, and particle physics to understand the behavior of particles under central forces. It is also used in the study of planetary orbits and in the design of spacecraft trajectories.

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