Exercise
https://texercises.com/exercise/repelling-particles/
Question
Solution
Short
Video
\(\LaTeX\)
No explanation / solution video to this exercise has yet been created.

Visit our YouTube-Channel to see solutions to other exercises.
Don't forget to subscribe to our channel, like the videos and leave comments!
Exercise:
Two positive particles with masses m_ and m_ and charges q_ and q_ are placed at an initial distance r_. abcliste abc Derive a formal expression for the two particles' final velocities v_ and v_. abc Calculate the velocities for a proton and an alpha particle with an initial distance rO. abcliste

Solution:
abcliste abc The total energy of the system of the two charged particles corresponds to the initial potential energy which is the electric potential energy of particle in the potential of particle or vice versa: sscEpot k_Cfracq_ q_r_ The potential energy is converted to kinetic energy. In the final state the following relation holds true: sscEkin+sscEkin sscEpot fracleftm_ v_^+m_ v_^right k_Cfracq_ q_r_labelconsE From the conservation of momentum we know that m_ v_ m_ v_. We can therefore express v_ as a function of v_: v_ fracm_m_ v_ labelconsP Using this in refconsE we find fracleftm_+m_fracm_^m_^right v_^ k_Cfracq_ q_r_ fracleftfracm_+m_m_right m_ v_^ k_Cfracq_ q_r_ Solving this for v_ leads to v_ vaF With refconsP we find the corresponding expression for v_: v_ fracm_m_vaF vbF abc For a proton m_ma q_qa and an alpha particle m_mb q_qb we can calculate the numerical values of v_ and v_: v_ sqrttimesnckctimesfracqatimesqbrfracmbmatimesma+mb v_ sqrttimesnckctimesfracqaXtimesqbXtimesnce^rfracmbXmaXtimesmaX+mbXtimesncu resultvaP v_ sqrttimesnckctimesfracqatimesqbrfracmambtimesma+mb v_ sqrttimesnckctimesfracqaXtimesqbXtimesnce^rfracmaXmbXtimesmaX+mbXtimesncu resultvbP abcliste
Meta Information
\(\LaTeX\)-Code
Exercise:
Two positive particles with masses m_ and m_ and charges q_ and q_ are placed at an initial distance r_. abcliste abc Derive a formal expression for the two particles' final velocities v_ and v_. abc Calculate the velocities for a proton and an alpha particle with an initial distance rO. abcliste

Solution:
abcliste abc The total energy of the system of the two charged particles corresponds to the initial potential energy which is the electric potential energy of particle in the potential of particle or vice versa: sscEpot k_Cfracq_ q_r_ The potential energy is converted to kinetic energy. In the final state the following relation holds true: sscEkin+sscEkin sscEpot fracleftm_ v_^+m_ v_^right k_Cfracq_ q_r_labelconsE From the conservation of momentum we know that m_ v_ m_ v_. We can therefore express v_ as a function of v_: v_ fracm_m_ v_ labelconsP Using this in refconsE we find fracleftm_+m_fracm_^m_^right v_^ k_Cfracq_ q_r_ fracleftfracm_+m_m_right m_ v_^ k_Cfracq_ q_r_ Solving this for v_ leads to v_ vaF With refconsP we find the corresponding expression for v_: v_ fracm_m_vaF vbF abc For a proton m_ma q_qa and an alpha particle m_mb q_qb we can calculate the numerical values of v_ and v_: v_ sqrttimesnckctimesfracqatimesqbrfracmbmatimesma+mb v_ sqrttimesnckctimesfracqaXtimesqbXtimesnce^rfracmbXmaXtimesmaX+mbXtimesncu resultvaP v_ sqrttimesnckctimesfracqatimesqbrfracmambtimesma+mb v_ sqrttimesnckctimesfracqaXtimesqbXtimesnce^rfracmaXmbXtimesmaX+mbXtimesncu resultvbP abcliste
Contained in these collections:

Attributes & Decorations
Branches
Electrostatics
Tags
conservation of momentum, electric potential, electric potential energy, momentum
Content image
Difficulty
(3, default)
Points
0 (default)
Language
ENG (English)
Type
Calculative / Quantity
Creator by
Decoration
File
Link