Conservation of Angular Momentum and Energy

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SUMMARY

The discussion focuses on the conservation of angular momentum and energy in the context of a comet's motion from perihelion to aphelion. The key equations utilized are angular momentum (L = mvr) and energy (E = 0.5mv^2 + U). The user attempts to derive the relationship between the comet's initial and final radial distances (r1 and r2) and velocities (v1 and v2) but encounters difficulties due to the absence of gravitational concepts in their coursework. The solution requires correctly applying these principles to establish the relationship between the variables.

PREREQUISITES
  • Understanding of angular momentum (L = mvr)
  • Familiarity with energy conservation principles (E = 0.5mv^2 + U)
  • Basic knowledge of gravitational force (F = GMm/r^2)
  • Experience with Mathematica for solving equations
NEXT STEPS
  • Study the principles of Newton's Gravitation and its applications in orbital mechanics
  • Learn how to apply conservation laws in celestial mechanics
  • Explore the use of Mathematica for solving complex physics equations
  • Investigate the relationship between potential energy and kinetic energy in gravitational fields
USEFUL FOR

Students studying physics, particularly those focusing on celestial mechanics, as well as educators looking to reinforce concepts of angular momentum and energy conservation in orbital dynamics.

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Homework Statement


We measure a comet at perihelion to have a radial distance r1 and a velocity v1. Find the radial distance and velocity when it reaches aphelion.

Homework Equations


L=mvr=I*\omega
E=.5mv^2+U

The Attempt at a Solution


My professor skipped the chapter on Newton's Gravitation, F=GMm/r^2, so I don't think it applies here.

The book is entirely symbolic--the answer expected should be in terms of v1 and r1.

I labeled the radial distance and velocity r2 and v2, respectively. Since L is constant, m*v1*r1 = m*v2*r2, or v1*r1=v2*r2. Of course, this equation only gives the unknown radial distance in terms of the unknown velocity, or vice versa.

I don't think U = -GMm/r would apply here, since, again, my professor skipped the chapter on gravitation. I tried solving using a second equality, .5m*v1^2-GMm/r1=.5m*v2^2-GMm/r2, but when I solved the system of equations with Mathematica, I got a gigantic mess for an answer.

I appreciate your time.
 
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I think both your equations correct, you just did something wrong with the calculations. :smile:

BTW, what's mathematica ? Are you using a http://en.wikipedia.org/wiki/Mathematica" to solve these equations ?
 
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