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The energy of the particle from Problem 2 is given by 1 E = M2 +U = (2) 2mr2 2a Show that the motion of the particle r(t) is given by parametric equations: 1 t = - -esin ξ), r = a(1 – e cos E), (3) where w = [a/(ma³)]/2 is the mean angular velocity of the orbital motion and & is a parameter

The energy of the particle from Problem 2 is given by 1 E = M2 +U = (2) 2mr2 2a Show that the motion of the particle r(t) is given by parametric equations: 1 t = - -esin ξ), r = a(1 – e cos E), (3) where w = [a/(ma³)]/2 is the mean angular velocity of the orbital motion and & is a parameter

Classical Dynamics of Particles and Systems
Classical Dynamics of Particles and Systems
5th Edition
ISBN:9780534408961
Author:Stephen T. Thornton, Jerry B. Marion
Publisher:Stephen T. Thornton, Jerry B. Marion
Chapter8: Central-force Motion
Section: Chapter Questions
Problem 8.35P
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The energy of the particle from Problem 2 is given by 1 E = +U = 2a (2) 2mr2 Show that the motion of the particle r(t) is given by parametric equations: 1 t = - -esin ξ), r = a(1 – e cos {), (3) where w = [a/(ma*)]1/2 is the mean angular velocity of the orbital motion and & is a parameter (which ranges from 0 to 27 for one revolution). The first relation, which gives (t), is called the Kepler equation. Substituting r(t) into the equation of path (1) gives o(t). Hint: Use equations (3) to calculate dr/d£ dt/dg dr dt and then substitute r and r into equation (2). Find r/a and wt for twenty values of E between 0 and 47, for some value of e between 0 and 1 of your choice. Graph these pairs: r/a on the vertical axis and wt on the horizontal axis. They form a curve which is a trochoid. Graph the same parametric function for e → 1. This curve is a cycloid.
Transcribed Image Text:The energy of the particle from Problem 2 is given by 1 E = +U = 2a (2) 2mr2 Show that the motion of the particle r(t) is given by parametric equations: 1 t = - -esin ξ), r = a(1 – e cos {), (3) where w = [a/(ma*)]1/2 is the mean angular velocity of the orbital motion and & is a parameter (which ranges from 0 to 27 for one revolution). The first relation, which gives (t), is called the Kepler equation. Substituting r(t) into the equation of path (1) gives o(t). Hint: Use equations (3) to calculate dr/d£ dt/dg dr dt and then substitute r and r into equation (2). Find r/a and wt for twenty values of E between 0 and 47, for some value of e between 0 and 1 of your choice. Graph these pairs: r/a on the vertical axis and wt on the horizontal axis. They form a curve which is a trochoid. Graph the same parametric function for e → 1. This curve is a cycloid.
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