Solving a Celestial Mechanics Task with Velocity Vector Scattering

In summary, the task is to find the change in kinetic energy of a particle that is scattered by an angle from its original direction. The change in kinetic energy is expressed in terms of the initial and final velocities relative to the planet.
  • #1
Lambda96
195
67
Homework Statement
Find the difference in kinetic energy when the velocities of the object and planet are antiparallel
Relevant Equations
none
Hi,

the task is as follows

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Bildschirmfoto 2022-11-15 um 20.19.25.png
Unfortunately, I am not getting anywhere at all with task c. I have now proceeded as follows:

I assume that the calculation takes place in the reference system of the sun. In the task the following is valid, $$\vec{v}_{si}=-s\vec{v}_p$$ I have now simply assumed that the planet is scattered the velocity vector by the angle $$\theta$$ so the following is valid for the final velocity

$$\vec{v}_{sf}=-s\vec{v}_p*cos(\theta)$$

After that I simply made the difference in the kinetic energy, i.e.$$E_{kin,f}-E_{kin,i}=\frac{1}{2}m\vec{v}_{sf}^2 - \frac{1}{2}m\vec{v}_{si}^2$$If I now insert $$\vec{v}_{sf}=-s\vec{v}_p*cos(\theta)$$ and $$\vec{v}_{si}=-s\vec{v}_p$$, I immediately see that I do not get the required equation, since I have terms like $$s^2$$ and $$cos(\theta)^2$$. But I also don't understand how to get this equation where $$s$$ and $$cos(\theta)$$ occur, you have to square the velocity for the kinetic energy.
 
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  • #2
Lambda96 said:
I assume that the calculation takes place in the reference system of the sun.
Part (b) is setting you up for part (c). Thus, you can express the change in KE of ##\mathrm{\mathring{A}}## relative to the sun in terms of the initial and final velocities of ##\mathrm{\mathring{A}}## relative to the planet P! This makes things easier because the initial and final velocities relative to P have a simple relationship (as shown in part (a)).

Lambda96 said:
In the task the following is valid, $$\vec{v}_{si}=-s\vec{v}_p$$
OK. Can you use this to find an expression for ##\vec{v}_{pi}## in terms of ##s## and ##\vec{v}_P##?

Lambda96 said:
I have now simply assumed that the planet is scattered the velocity vector by the angle $$\theta$$ so the following is valid for the final velocity

$$\vec{v}_{sf}=-s\vec{v}_p*cos(\theta)$$
This equation can't be right. The right-hand side represents a vector quantity that has a direction either parallel or antiparallel to ##\vec{v}_P## (depending on the sign of ##\cos \theta##). But that's certainly not the right direction for ##\vec{v}_{sf}##. However, if you use the expression for the change in KE as given in part (b), you don't need ##\vec{v}_{sf}##. Instead, you need ##\vec{v}_{pf}##. Or, more specifically, you need ##\vec{v}_{pf} \cdot \vec{v}_P##.

Note that ##\mathrm{\mathring{A}}## is scattered by the angle ##\theta## in the frame of reference of the planet, not in the frame of reference of the sun.
 
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  • #3
Thanks for your help TSny 👍, with your help I was able to solve the task of 🙂
 
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FAQ: Solving a Celestial Mechanics Task with Velocity Vector Scattering

1. What is celestial mechanics?

Celestial mechanics is the branch of astronomy that studies the motion and behavior of celestial bodies, such as planets, moons, stars, and galaxies. It involves using mathematical equations and principles to understand the structure and dynamics of the universe.

2. What is a velocity vector scattering task?

A velocity vector scattering task is a problem in celestial mechanics where the goal is to determine the trajectory of a celestial body by analyzing its velocity vector, which is the speed and direction of its motion.

3. How is velocity vector scattering used to solve celestial mechanics tasks?

Velocity vector scattering is used to solve celestial mechanics tasks by applying mathematical equations and principles, such as Newton's laws of motion and the law of gravitation, to analyze the motion of celestial bodies and predict their future positions and behaviors.

4. What are the challenges of solving a celestial mechanics task with velocity vector scattering?

One of the main challenges of solving a celestial mechanics task with velocity vector scattering is the complexity of the equations and calculations involved. It also requires a deep understanding of physics and mathematics, as well as advanced computational tools and techniques.

5. How can solving a celestial mechanics task with velocity vector scattering benefit society?

Solving celestial mechanics tasks with velocity vector scattering can help us better understand the universe and the forces that govern its behavior. This knowledge can be applied to various fields, such as space exploration, satellite navigation, and understanding climate change on Earth. It can also lead to advancements in technology and a deeper appreciation for the wonders of our universe.

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