Distance of Closest Approach of Particle to PLanet

This will be the distance of closest approach.In summary, the question asks for the distance of closest approach of a particle with unknown mass and initial velocity v0, approaching a planet with mass M from a far distance. The relevant equation is Ueff(r) = (angular momentum)2/2mr2 + U(r) and the impact parameter b is not relevant since the particle does not impact the planet. To calculate the distance of closest approach, the angular momentum L of the particle must be calculated using the initial distance and speed. Energy considerations can then be used to find the minimum value of r, which will be the distance of closest approach.
  • #1
mm8070
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Homework Statement


A particle, unknown mass, has velocity v0 and impact parameter b. It goes towards a planet, mass M, from very far away. Find from scratch (? I'm not sure why it says from scratch), the distance of closest approach.


Homework Equations


I believe this equation is relevant: Veff(r)=L2/2mr + V(r)


The Attempt at a Solution


I haven't attempted this problem because I have no idea what distance of closest approach is. I looked throughout my book and haven't found anything.
 
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  • #2
This question really doesn't make sense to me. The "distance of closest approach" is just what it says- the distance a which the particle is closest to the planet as it flies by. Of course, it it hit the planet, that would be 0. But to calculate such a thing you would have to compute its trajectory which would involve knowing not only its initial distance and speed but also it initial direction of travel. when you said "velocity [itex]v_0[/itex], is that a velocity vector? That would help buit then your formula would be adding a number ([itex]L^2/2mr[/itex]) to a vector (V(r)). In any case, I don't see how the "impact parameter" would be relevant if the particle does not "impact" the planet.
 
  • #3
Maybe i should have written the equation as Ueff(r) = (angular momentum)2/2mr2 + U(r). Where U(r) is the potential energy. I also should have mentioned that part b says to use the section in my book about hyperbolas to show that the distance of closest approach is k/(ε + 1) where k and ε are some ridiculous constants that I'm certain would waste your time if I gave them to you. I'm sorry about that :frown:
 
  • #4
Draw a line through the center of the planet, parallel to v0. The particle is a distance b from this line initially. Use this information to calculate the angular momentum L of the particle.

Once you have that, you can use energy considerations to figure out what the minimum value of r the particle can achieve is.
 
  • #5


I would suggest approaching this problem by first understanding the concept of distance of closest approach. This refers to the minimum distance between the particle and the planet during their interaction. In other words, it is the point at which their paths are closest to each other.

To solve for this distance, we can use the equation provided, Veff(r)=L2/2mr + V(r), where Veff(r) is the effective potential energy, L is the angular momentum, m is the mass of the particle, r is the distance between the particle and the planet, and V(r) is the potential energy between the two bodies.

To find the distance of closest approach, we can set the effective potential energy equal to zero, as this represents the point of minimum potential energy. Then, we can solve for r to find the distance of closest approach.

It is also important to take into account the initial conditions given in the problem, such as the velocity and impact parameter of the particle. These will affect the path and therefore the distance of closest approach.

In summary, to find the distance of closest approach of a particle to a planet, we can use the equation Veff(r)=L2/2mr + V(r) and set it equal to zero, taking into account the initial conditions. This will give us the minimum distance between the two bodies during their interaction.
 

1. What is the distance of closest approach of a particle to a planet?

The distance of closest approach of a particle to a planet is the shortest distance between the particle and the planet's surface. It is measured in meters and is an important factor in understanding the gravitational interactions between the particle and the planet.

2. How is the distance of closest approach calculated?

The distance of closest approach is calculated using the laws of gravity, specifically the equation for gravitational force. This takes into account the masses of the particle and the planet, as well as the distance between them.

3. Why is the distance of closest approach important?

The distance of closest approach is important because it helps determine the strength of the gravitational force between the particle and the planet. It also plays a role in predicting the trajectory of the particle and its potential impact on the planet.

4. Can the distance of closest approach change?

Yes, the distance of closest approach can change. It is affected by factors such as the velocity and direction of the particle, as well as the gravitational forces of other bodies in the vicinity. It can also change over time as the particle's trajectory is altered.

5. How does the distance of closest approach relate to the orbital path of the particle?

The distance of closest approach is directly related to the orbital path of the particle. It is the point at which the particle is closest to the planet, and from there, it either continues on its original trajectory or is affected by the planet's gravitational pull and may enter into an orbit around the planet.

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